➕ Algebra 1 · Systems of Equations
Solving Systems by Elimination: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Comparing receipts
You and a friend each order at the same taco stand, and both orders get the same prices. Your receipt shows 4 tacos and 3 drinks for 18 dollars. Your friend's shows 4 tacos and 1 drink for 14 dollars. Both orders have the same 4 tacos, so the difference between the receipts comes from the drinks alone: your order has 2 more drinks and costs 4 dollars more, so a drink costs 2 dollars. Then 4 tacos cost 14 − 2 = 12 dollars, so a taco costs 3 dollars.
How it connects: Comparing the receipts is subtracting equations: (4t + 3d) − (4t + d) = 18 − 14 gives 2d = 4, so d = 2. It works because the taco terms match and cancel. If your friend had ordered 2 tacos and 1 drink for 8 dollars, the taco terms would not match, so you would double that whole receipt first: 4 tacos and 2 drinks for 16 dollars. The limit of the picture: it only works because both receipts use the same prices. A coupon on one receipt would break it.
Doubling the recipe
A trail-mix recipe says a batch has n cups of nuts and r cups of raisins and makes 3 cups in all, so n + r = 3. Two batches hold twice the nuts, twice the raisins and twice the total: 2n + 2r = 6. Nobody doubles the nuts and the raisins and still says the total is 3 cups.
How it connects: Multiplying an equation by a number multiplies every term, including the constant on the right: n + r = 3 becomes 2n + 2r = 6, not 2n + 2r = 3. The new equation is true for exactly the same n and r as the old one, which is why multiplying does not change the solution. The limit: the multiplier must not be zero (zero batches would turn the equation into 0 = 0, which says nothing). And when you multiply one equation of a system, the other equation stays as it is.
Cancel each other out
When two forces or effects are equal and opposite, people say they cancel each other out. If one effect adds 5 and the other subtracts 5, the net effect is 0.
How it connects: Opposite coefficients cancel when you add: 4y + (−4y) = 0, so you add the equations when a variable has opposite coefficients. The limit: only exact opposites of the same kind of term cancel. 4y and −4x are different kinds of terms, and 4y + (−3y) leaves y. If the coefficients are equal, like 4y and 4y, adding gives 8y, so subtract instead.
Two heads are better than one
The proverb says that two people thinking together can solve a problem that one person alone might not.
How it connects: One equation with two unknowns, like x + y = 9, has many solutions: (6, 3), (0, 9), (−2, 11) and more. A second equation is the second clue that can narrow them down to one pair. The limit: two equations do not always give exactly one solution. If the two equations contradict each other there is no solution, and if they say the same thing twice there are infinitely many.
One contestant leaves each round
In many cooking competitions, one contestant is eliminated each round, and the rest keep competing.
How it connects: In the elimination method, you eliminate one variable so you can solve for the other. The limit: the eliminated variable is not gone for good, unlike a contestant. After you find x, you go back for y, and the answer is the pair (x, y). Reporting only x leaves out half the solution.
Row and seat
A stadium ticket might say row 12, seat 8. Row 12 alone does not tell you where to sit, and neither does seat 8 alone. You need both, in the right order: row 12, seat 8 is a different seat from row 8, seat 12.
How it connects: A solution of a system is an ordered pair (x, y), and the order matters: (2, 5) is not (5, 2). Finding x is like finding the row, and you still have to find the seat. The limit: seat numbers are positive whole numbers, but a solution of a system can have negative or fractional coordinates, like (−2, 1/2).
The group project
Why did 3y and −3y skip the group project? They knew that together they would add up to nothing.
How it connects: Opposite coefficients add to 0, so adding the equations eliminates that variable. The limit: the variable is not gone from the answer. After you find the other variable, back-substitute to find this one too. Equal coefficients like 3y and 3y do not cancel when you add; subtract them instead.
Memory help
LAMBS
Line up the equations in standard form. Adjust by multiplying (every term) until one variable has opposite or equal coefficients. Merge: add if the coefficients are opposites, subtract if they are equal, then solve. Back-substitute to find the other variable. Show it works by checking both original equations. Skip any letter that does not apply: if the coefficients already match, there is nothing to Adjust.
Opposites add
Opposites add, twins subtract; neither one? Multiply first, that is a fact. (“Twins” means the very same coefficient, like 3y and 3y. “Opposites” means like 3y and −3y. Two coefficients with the same sign but different sizes, like 3y and 5y, are neither one.)
Stack it like an addition problem
Write the equations one above the other with x over x, y over y, = over =, and constants over constants. Circle the pair of terms you want to cancel. Draw a line underneath and add (or subtract) column by column.
The minus reaches everyone
Subtracting the equation 3x − 2y = 10 means subtracting 3x, subtracting −2y (which adds 2y) and subtracting 10. The minus sign reaches every term, including the constant on the right.
The constant gets multiplied too
Multiplying x − 3y = 6 by 4 gives 4x − 12y = 24. The 6 is a term like any other. If you leave it as 6, you have changed the equation.
x alone is half an answer
Elimination finds one variable first. The solution of a system is an ordered pair, so go back and find the other variable, then check both numbers in both equations.