➕ Algebra 1 · Systems of Equations

Solving Systems by Graphing

Graph two linear equations, read the crossing point as the solution, check it in both equations, and spot no solution or infinitely many solutions.

Grades 9, 10⏱ 15 min✔ AI-checkedCCSS 8.EE.C.8aCCSS A-REI.C.6CCSS A-REI.D.11

About standards and checking

Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Three blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-07.

8.EE.C.8a: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
A-REI.C.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
A-REI.D.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations.

By the end you can…

Explain that the solution of a system of two linear equations is the ordered pair (x, y) that makes both equations true, and that it is the point where the two lines cross.
Graph both lines (from slope-intercept form, or after rewriting an equation) and read the crossing point.
Check a possible solution in both original equations, and tell when a graph gives only an estimate.
Decide from the graph or from the slopes whether a system has one solution, no solution or infinitely many solutions.

Lesson (school style)

A system of equations is a set of equations about the same unknowns that must all be true at the same time. In this lesson a system is two linear equations with the same two variables, x and y. One equation alone is satisfied by a whole line of pairs (x, y). A system asks for a pair that works in both equations. In the lesson on graphing linear equations you learned that the graph of an equation is the picture of all its solutions. So graph both lines and look at where they meet: a point on both lines makes both equations true. Sometimes the lines cross once (one solution), sometimes they never meet (no solution), and sometimes they are the same line (infinitely many solutions).

What a system is, and what its solution is

A system of equations is two or more equations that use the same variables and must be true together. Here is one:

y = x + 2
y = −3x + 6

A solution of a system is an ordered pair (x, y) that makes both equations true at the same time. Test two pairs:

(1, 3): in the first equation, 3 = 1 + 2 is true. In the second, 3 = −3(1) + 6 is true. Both are true, so (1, 3) is the solution of the system.
(2, 4): in the first equation, 4 = 2 + 2 is true. In the second, −3(2) + 6 = 0 and 4 ≠ 0, so it is false. The pair fits only one equation, so it is not a solution of the system.

Why the crossing point is the solution

From the lesson on graphing linear equations: every point on a line is a solution of its equation, and every solution of the equation is a point on the line. So:

A point on the first line is a solution of the first equation.
A point on the second line is a solution of the second equation.
A point on both lines is a solution of both equations. That is exactly a solution of the system.

A point on both lines is a point where the lines meet, so the solutions of the system are the points the two graphs have in common. For the system above, the lines cross at (1, 3), so the solution is (1, 3).

The method: graph both, read the crossing, check

Get both equations ready to graph. Slope-intercept form, y = mx + b, is the easiest. If an equation is not solved for y, solve it for y first (or use its intercepts, as in the graphing lesson).
Graph both lines on the same axes. For each line, plot the point (0, b), then use the slope (rise ÷ run) to find more points. Use a ruler and extend each line past your points.
Find where the lines cross and read the coordinates of that point: x first, then y.
Check the point in both original equations.
Write the solution as an ordered pair (x, y).

Use one set of axes and one scale for both lines, and write each line's equation next to it so you do not mix them up.

Checking: why both equations?

A point on only one line fits only one equation, so test a possible solution in both. The check also catches reading errors. You might read a crossing as (2, 5) when the lines really cross slightly off that spot, and the check will show it.

If both equations are true, you found the solution. If either one is false, that pair is not the solution: go back and re-read the graph.

A solution has two numbers. Writing "x = 1" alone is half an answer. Write the pair, like (1, 3), with x first.

Exact answer or estimate?

Graphing gives an exact solution only when the lines cross at a point you can read exactly, such as a grid point like (1, 3), and the check works in both equations.

When the lines cross between grid lines, you can only estimate the point, for example “about (1.3, 1.7).” An estimate is not the solution. It can fail the check by a little, because the exact coordinates may be numbers like 4/3 that fall between tick marks. A drawn line also has thickness, so a crossing at (2, 3) and one at (2.1, 3) can look the same.

So a clean crossing plus a passing check means an exact answer. A crossing between grid lines means an estimate: use it as a guide, then find the exact answer with algebra (the next lessons).

One solution, no solution, infinitely many solutions

Two lines in a plane can relate in just three ways, and that tells you how many solutions the system has. To compare non-vertical lines, write both equations as y = mx + b.

Different slopes. The lines cross at exactly one point, so the system has one solution.
Same slope, different y-intercepts. The lines are parallel. They never meet, so the system has no solution. For example, y = (1/2)x + 5 and y = (1/2)x − 3 both rise 1 for every 2 steps right, but one has b = 5 and the other has b = −3.
Same slope and same y-intercept. Both equations describe the same line, so every point on it is a solution: infinitely many solutions. The equations can look different. For example, y = 2x + 3 and 4x − 2y = −6 are the same line, because solving the second for y gives y = 2x + 3. Its solutions include (0, 3), (1, 5) and (−2, −1).

“Infinitely many” is not one special number to plug in. It means every point on the line works.

The same slope does not decide it by itself. After you see equal slopes, compare the y-intercepts. Also, if two lines look as if they do not meet on your graph, compare the slopes before you say “no solution.” Lines with different slopes always cross somewhere, even if that is off the edge of your grid.

Vertical lines x = a have no slope-intercept form, but the same three outcomes hold. A vertical line and a non-vertical line cross exactly once. Two different vertical lines, like x = 1 and x = 4, are parallel (no solution). Two equations for the same vertical line have infinitely many solutions.

The x-coordinate and equations with variables on both sides

At the crossing point of y = x + 2 and y = −3x + 6, the two lines have the same y-value. So the x-coordinate of the crossing is the number that makes the two right sides equal. It solves x + 2 = −3x + 6, an equation with variables on both sides.

Solve it: add 3x to both sides to get 4x + 2 = 6, subtract 2 to get 4x = 4, so x = 1. Put x = 1 into either line to find y = 3. That matches the crossing (1, 3) on the graph.

It also works the other way. To solve a one-variable equation like x + 2 = −3x + 6 by graphing, graph each side as its own line, y = x + 2 and y = −3x + 6, and read the x-coordinate where they cross. If the crossing is between grid lines, you get only an approximate x, so test it in the equation.

Key words

System of equations: Two or more equations that use the same variables and must be true at the same time. In this lesson, two linear equations in x and y.
Solution of a system: An ordered pair (x, y) that makes both equations true at the same time. On the graph, it is a point on both lines.
Point of intersection: A point where two graphs meet. For two lines in a system, its coordinates are a solution of the system.
Parallel lines: Different lines in a plane that never meet. For non-vertical lines, that means the same slope and different y-intercepts.
No solution: No ordered pair makes both equations true. On the graph, the lines are parallel.
Infinitely many solutions: Every point on the shared line is a solution. This happens when both equations describe the same line.

Worked examples

Example 1. Solve the system y = x + 2 and y = −3x + 6 by graphing.

Both equations are already in slope-intercept form. The first has b = 2 and slope 1. The second has b = 6 and slope −3.
Graph the first: plot (0, 2), then go up 1 and right 1 to (1, 3) and again to (2, 4). Draw the line through them.
Graph the second on the same axes: plot (0, 6). The slope −3 is −3/1, so go down 3 and right 1 to (1, 3) and again to (2, 0). Draw the line through them.
Both lines pass through (1, 3). That is where they cross.
Check in both equations. First: 3 = 1 + 2, true. Second: 3 = −3(1) + 6, true.
The solution is (1, 3), which means x = 1 and y = 3.

Answer: Answer: (1, 3)

Example 2. Solve x + 3y = 3 and x − y = −5 by graphing.

Neither equation is solved for y, so rewrite each. First: subtract x from both sides to get 3y = −x + 3, then divide every term by 3: y = −(1/3)x + 1. So b = 1 and the slope is −1/3.
Second: subtract x from both sides to get −y = −x − 5, then divide every term by −1: y = x + 5. So b = 5 and the slope is 1.
Graph the first line: plot (0, 1). The slope −1/3 means down 1 and right 3, which gives (3, 0). Going the other way, up 1 and left 3, gives (−3, 2).
Graph the second line: plot (0, 5). The slope is 1, so going down 1 and left 1 gives (−1, 4), then (−2, 3), then (−3, 2).
Both lines pass through (−3, 2), so that is the crossing.
Check in the original equations. First: −3 + 3(2) = −3 + 6 = 3, true. Second: −3 − 2 = −5, true.

Answer: Answer: (−3, 2)

Example 3. How many solutions does the system y = (3/2)x − 2 and 3x − 2y = 8 have?

Write the second equation in slope-intercept form. Subtract 3x from both sides: −2y = −3x + 8. Divide every term by −2: y = (3/2)x − 4.
Compare the two lines. The first has slope 3/2 and b = −2. The second has slope 3/2 and b = −4.
The slopes are the same and the y-intercepts are different, so the lines are parallel. On a graph, both rise 3 for every 2 steps right, but one crosses the y-axis at (0, −2) and the other at (0, −4). They never meet.
Another way to see it: multiply the first equation by 2 to get 2y = 3x − 4, which gives 3x − 2y = 4. A pair (x, y) cannot make 3x − 2y equal 4 and also 8.
No point is on both lines, so no ordered pair makes both equations true.

Answer: Answer: No solution (parallel lines)

Example 4. How many solutions does the system y = 3x − 1 and 9x − 3y = 3 have?

Write the second equation in slope-intercept form. Subtract 9x from both sides: −3y = −9x + 3. Divide every term by −3: y = 3x − 1.
Both equations are y = 3x − 1: the same slope, 3, and the same b, −1. They describe the same line.
Every point on that line is a solution of both equations. Try three points: (0, −1), (1, 2) and (−2, −7). In the second equation: 9(0) − 3(−1) = 3, 9(1) − 3(2) = 3, and 9(−2) − 3(−7) = −18 + 21 = 3. All true. (They also fit the first equation, since each satisfies y = 3x − 1.)
The system has infinitely many solutions: every ordered pair on the line y = 3x − 1.

Answer: Answer: Infinitely many solutions (every point on the line y = 3x − 1)

Example 5. Solve y = 2x − 1 and y = −x + 3 by graphing. What does the graph tell you, and what does it not?

Graph the first: plot (0, −1), then up 2 and right 1 to (1, 1) and (2, 3).
Graph the second: plot (0, 3), then down 1 and right 1 to (1, 2), (2, 1) and (3, 0).
At x = 1 the first line is at height 1 and the second is at height 2, so the first is below. At x = 2 the first is at height 3 and the second is at height 1, so the first is above. The lines swapped places between x = 1 and x = 2, so they cross there, between grid lines.
Reading the graph, the crossing looks like about (1.3, 1.7). That is an estimate, not a solution.
Test the estimate. In the first equation with x = 1.3: 2(1.3) − 1 = 1.6, not 1.7. The estimate does not make the first equation exactly true.
For the exact answer, set the right sides equal: 2x − 1 = −x + 3, so 3x = 4 and x = 4/3. Then y = 2(4/3) − 1 = 5/3. Check in the second equation: −4/3 + 3 = −4/3 + 9/3 = 5/3. True.
The exact solution is (4/3, 5/3). As decimals, 4/3 is about 1.33 and 5/3 is about 1.67. The graph found the right neighborhood, but only the check and the algebra give the exact pair.

Answer: Answer: The lines cross between grid points. The exact solution is (4/3, 5/3). Reading the graph gives only an estimate, about (1.3, 1.7).

Example 6. Climbing gym A charges a 12-dollar membership fee plus 2 dollars per visit. Gym B has no membership fee and charges 4 dollars per visit. When is the total cost the same? Solve by graphing.

Let x be the number of visits and y the total cost in dollars. Gym A: y = 2x + 12. Gym B: y = 4x.
Gym A's line starts at (0, 12) and rises 2 for each visit. Gym B's line starts at (0, 0) and rises 4 for each visit. Use a y-axis that goes up to at least 30.
Gym B starts lower but climbs faster, so its line catches up. The lines cross at (6, 24).
Check. Gym A: 2(6) + 12 = 24. Gym B: 4(6) = 24. Both are true.
The crossing means: at 6 visits the two gyms cost the same, 24 dollars. With fewer than 6 visits, B costs less (at 2 visits, 8 dollars against 16). With more than 6, A costs less (at 10 visits, 32 dollars against 40).
Only whole numbers of visits make sense, and x cannot be negative, so only part of each line fits the story.

Answer: Answer: (6, 24): at 6 visits both gyms cost 24 dollars.

Example 7. Solve x = −2 and y = 3x + 1 by graphing.

The graph of x = −2 is a vertical line through (−2, 0). Every point on it has x-coordinate −2.
Graph y = 3x + 1: plot (0, 1), then up 3 and right 1 to (1, 4). Going the other way, down 3 and left 1 gives (−1, −2), then (−2, −5).
The vertical line x = −2 passes through (−2, −5), which is on the slanted line. That is the crossing.
Check. x = −2: the x-coordinate is −2, true. y = 3x + 1: 3(−2) + 1 = −5, true.

Answer: Answer: (−2, −5)

In plain English

A system is two clues about the same two mystery numbers, x and y. Both clues must be true at once.

One clue alone fits many pairs. On a graph, one clue is a whole line of pairs. A pair that fits both clues has to sit on both lines, so it is where the lines cross.

So you draw both lines and find the crossing. You read its x and y. Then you test the pair in both clues. If both work, you are done.

Lines can also miss each other. Side-by-side lines never cross, so no pair fits both clues. Or the two clues may be the same line in disguise, and then every point on it fits.

One more thing. A graph is only as exact as your drawing. If the lines cross between the grid marks, you only get an estimate. Test it, and use algebra when you need the exact answer.

See it

The lines y = x + 2 and y = −3x + 6 cross at (1, 3). That pair makes both equations true, so it is the solution.

Compare the slopes, then the y-intercepts. Different slopes cross once. Parallel lines never meet. The same line shares every point.

The same idea, the fun way

Pick the one that clicks. Each one explains the same concept from a different angle.

Relationships

Meet at the corner

Jo and Kai are meeting downtown. On a city map with an x-axis running east and a y-axis running north, Jo texts, “I’ll be on Main Street,” which is the vertical line x = 3. Kai texts, “I’ll be on 5th Avenue,” which is the horizontal line y = 5. Only one spot is on both streets: the corner (3, 5). If Kai stands at (8, 5), that is on 5th Avenue but not on Main Street, so the two have not met.

How it connects: Each text is an equation, and a spot that satisfies both is the solution of the system. The spot (8, 5) satisfies only one equation, which is why you check a possible solution in both. The limit of the picture: streets at right angles are a special case. Slanted lines, like y = 2x + 1, also cross in exactly one point whenever their slopes are different. Two different streets that run the same direction never meet, which is the no-solution case.

Food

Two receipts, two unknown prices

At a food truck, let x be the price of one taco and y the price of one drink, both in dollars. Receipt 1 says 2 tacos and 1 drink cost 8 dollars: 2x + y = 8. Receipt 2 says 1 taco and 2 drinks cost 7 dollars: x + 2y = 7. Receipt 1 alone fits many price pairs, such as (1, 6) and (2, 4), but neither fits Receipt 2, since 1 + 2(6) = 13 and 2 + 2(4) = 10. Only one pair fits both receipts: tacos cost 3 dollars and drinks cost 2 dollars, the point (3, 2).

How it connects: Each receipt is a line (all the price pairs that fit it), and the real prices are where the two lines cross. The limit of the picture: prices cannot be negative, so only part of each line is used. And if Receipt 2 were just Receipt 1 doubled (4 tacos and 2 drinks for 16 dollars), it would be the same line and add no new information, so infinitely many price pairs would fit both.

Gaming

The two-lock vault

In a made-up video game, a vault has two locks and two dials, x and y. Lock A opens whenever the dials satisfy y = x + 4. Lock B opens whenever they satisfy y = −x + 12. The vault door opens only when both locks are open. Try the dials (1, 5): Lock A opens because 5 = 1 + 4, but Lock B stays shut because −1 + 12 = 11, not 5. Try (4, 8): 8 = 4 + 4 and 8 = −4 + 12, so both locks open and the door unlocks.

How it connects: Each lock is one equation. A dial setting on only one line, like (1, 5), opens only one lock, so a possible solution must be tested in both equations. The limit of the picture: it is a made-up game. A system whose lines are parallel is a vault that can never open (no solution). A system that gives the same line twice is a vault where every setting that opens one lock also opens the other (infinitely many solutions).

Sports

Lane lines in a pool

Look at the black lane lines painted on the bottom of a swimming pool. They run in the same direction, side by side, and stay the same distance apart. Follow them as far as you like: they never meet.

How it connects: Same direction means same slope. Staying apart means different y-intercepts. Parallel lines never meet, so a system whose lines are parallel has no solution. The limit of the picture: a pool’s lane lines are short segments, while a line in math goes on forever. And two lines with the same slope and the same y-intercept are one line, like painting a line on top of itself. They share every point, so that system has infinitely many solutions.

Proverb

Look before you leap

The proverb advises you to think about what you are doing before you jump into it.

How it connects: Before you announce a crossing point, look: substitute it into both equations. A graph can be slightly off, and a point that fits only one equation is not a solution of the system. The limit of the proverb: it is general advice about caution. Here it means one specific action, testing the pair in both original equations.

Everyday joke

Parallel lines at the cafe

Two parallel lines sat down at a cafe. “We have so much in common,” said one. “Same slope, same direction.” “True,” said the other, “but we will never meet.”

How it connects: Parallel lines have the same slope and different y-intercepts, so they never meet: the system has no solution. The limit of the joke: having the same slope is not enough to guarantee no solution. If the y-intercepts match too, the two lines are one line, and they share every point.

Memory help

Acronym

GRID

Get each equation ready to graph (usually y = mx + b). Ruler-draw both lines on one grid (plot (0, b), then use the slope). Identify the crossing point and read (x, y). Double-check it in both equations.

Rhyme

Cross, parallel, same

“Cross once, one solution found. Parallel lines, no common ground. Same line twice, solutions abound.” This is for lines you can write as y = mx + b: crossing means different slopes, parallel means the same slope with different b, and the same line means the same slope and the same b.

Mnemonic

Slope first, then b

Compare the slopes first. Different slopes: the lines cross once, so there is one solution. Same slopes: now look at b. Different b means side by side (no solution). The same b means one line drawn twice (infinitely many solutions).

Trap alert

A solution has two numbers

The solution of a system is a pair (x, y). Saying “x = 1” alone is half an answer. And (3, 1) is a different point from (1, 3), because x always comes first.

Trap alert

One line is not enough

A point on only one line makes only one equation true. Test every possible solution in both equations before you write it down.

Picture it

Circle, label, check

Draw the two lines in two colors and write each equation next to its line. Circle the crossing point and write its coordinates beside the circle. Then write two short lines underneath: “Equation 1: true” and “Equation 2: true.”

Study notes

Key ideas

A system of two linear equations asks for an ordered pair (x, y) that makes both equations true. That pair is the solution of the system.
Each line is the set of solutions of one equation. A point on both lines is a solution of both, so the solution is where the lines cross.
To solve by graphing: write each equation as y = mx + b, graph both lines on the same axes, read the crossing, and check it in both equations.
For non-vertical lines: different slopes give one solution. The same slope with different y-intercepts (parallel lines) gives no solution. The same slope and the same y-intercept (the same line) gives infinitely many solutions.
Graphing is exact only when the crossing is a point you can read exactly and the check works. Otherwise it is an estimate.

Steps

Write each equation as y = mx + b (solve for y if you need to).
Graph both lines on the same axes with one scale: plot (0, b), then use the slope.
Find where the lines cross and read (x, y).
Check the point in both original equations.
Write the solution as an ordered pair, or say no solution or infinitely many solutions.

Formulas

Solution of a system: An ordered pair (x, y) that makes both equations true
One solution: m₁ ≠ m₂ (non-vertical lines with different slopes): the lines cross exactly once
No solution: m₁ = m₂ and b₁ ≠ b₂: parallel lines
Infinitely many solutions: m₁ = m₂ and b₁ = b₂: the same line
x-coordinate of the crossing: Solve m₁x + b₁ = m₂x + b₂ for x, then find y

Watch out

Test the point in both equations. A point on only one line fits only one equation.
Give both coordinates as an ordered pair, x first: (1, 3), not just x = 1.
The same slope does not by itself mean no solution. Compare the y-intercepts: different means parallel (no solution), the same means one line (infinitely many).
If the lines do not meet on your grid, compare the slopes before you say “no solution.” Lines with different slopes cross somewhere, maybe off the edge of your graph.
A crossing between grid lines is only an estimate. Check it, and use algebra for the exact answer.
When you solve an equation for y, divide every term, and keep the signs right when you divide by a negative number.

Why it works

Why is the crossing point the solution? Each line is the set of solutions of one equation. A point on both lines is in both sets, so it makes both equations true. And any solution of the system must be on both lines.
Why do lines with different slopes cross exactly once? Write them as y = m₁x + b₁ and y = m₂x + b₂. The vertical gap between them is (m₁ − m₂)x + (b₁ − b₂). If the slopes differ, this gap changes by the same nonzero amount for each 1 step in x, so it is zero at exactly one value of x. If the slopes are equal, the gap is the same everywhere: never zero (parallel) or always zero (the same line).
Why does the x-coordinate of the crossing solve m₁x + b₁ = m₂x + b₂? At the crossing both lines have the same y, so the two expressions for y are equal there.

Common mistakes

✗ MistakeThe solution of a system is the x-value, like x = 2.

✓ InsteadThe solution is an ordered pair (x, y) that makes both equations true. Both numbers are needed.

✗ MistakeIf a point is on one of the lines, it is a solution of the system.

✓ InsteadIt must be on both lines, so it must make both equations true. Check it in both.

✗ MistakeTwo lines with the same slope have no solution.

✓ InsteadThat is true only if the y-intercepts differ (parallel lines). With the same slope and the same y-intercept, the lines are the same, and there are infinitely many solutions.

✗ MistakeIf the lines do not cross on my graph, the system has no solution.

✓ InsteadLines with different slopes always cross somewhere, even if that is off the edge of the graph. Compare the slopes to decide.

✗ MistakeA graph always gives the exact answer, so I do not need to check.

✓ InsteadA reading from a graph can be slightly off, and a crossing between grid lines can only be estimated. Check it in both equations.

Teach it back

Explain to a friend how to solve the system y = −3x + 10 and y = x − 2 by graphing, and how to be sure the answer is right.

Explaining it out loud shows what you really know. Did you…

I graphed the first line from (0, 10) with slope −3 (down 3, right 1) and the second from (0, −2) with slope 1 (up 1, right 1).
I read the crossing point (3, 1), with x first and then y.
I checked it in both equations: −3(3) + 10 = 1 and 3 − 2 = 1.
I said that the slopes (−3 and 1) are different, so there is exactly one solution, and that parallel lines would mean no solution while the same line would mean infinitely many.

Your 3-week memory plan

TodayRead the lesson, then try the flashcards.
TomorrowRecall it from memory first, then check.
In 3 daysTake the true/false and practice questions.
In 1 weekMix it with older topics and teach it back.
In 3 weeksA last quick review before it fades.

Downloads a calendar file (.ics). Works with Google, Apple and Outlook calendars. Nothing is sent to us.