➕ Algebra 1 · Systems of Equations
Solving Systems by Substitution: 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.
Sticks of butter
In the United States, one stick of butter is 8 tablespoons. A recipe asks for 2 sticks plus 4 tablespoons of butter, but you only own a tablespoon measure. Let s be the number of tablespoons in one stick, so s = 8. The recipe needs 2s + 4 tablespoons. Swap in what s equals, in parentheses, and you get 2(8) + 4 = 20 tablespoons.
How it connects: The fact s = 8 lets you write 8 anywhere s appears, just as y = 5x + 1 lets you write (5x + 1) anywhere y appears. Notice 2(8), not 28: the parentheses show that the 2 multiplies the swapped-in amount. The limit of the picture: the swap is allowed only because a stick really equals 8 tablespoons. You cannot swap in something that is merely close. And in a system, you use one equation to get the swap and the other equation to put it into.
Two clues about two ages
Jo is 4 years older than Sam, and together their ages add up to 30. Let j be Jo’s age and s be Sam’s age, both in years. Clue 1 says j = s + 4. Clue 2 says j + s = 30. Use clue 1 to rewrite Jo’s age inside clue 2: (s + 4) + s = 30, so 2s + 4 = 30, then 2s = 26 and s = 13. Then j = 13 + 4 = 17.
How it connects: Clue 1 is the swap and clue 2 is where it goes. Only s is left, so you can solve, and then back-substitute to get j. Check both clues: 17 = 13 + 4 and 17 + 13 = 30. The limit of the picture: both clues must describe the same moment. If one clue said “in 5 years,” you would have to write it in terms of today’s ages first. And you never put clue 1 back into clue 1.
Counting baskets
In one game, a team made 20 baskets worth 47 points in all. Every basket was worth 2 or 3 points (ignore free throws). Let a be the number of 2-point baskets and b be the number of 3-point baskets. The basket count gives a + b = 20, so a = 20 − b. The points give 2a + 3b = 47. Swap in for a: 2(20 − b) + 3b = 47. Distribute: 40 − 2b + 3b = 47, so b = 7 and a = 13.
How it connects: The parentheses show that each of the (20 − b) two-point baskets is worth 2 points: 2(20 − b), not 2 · 20 − b. Check: 13 + 7 = 20 baskets, and 2(13) + 3(7) = 26 + 21 = 47 points. The limit of the picture: counts must be whole numbers that are not negative, and real games also have free throws worth 1 point each, which this example ignores.
The cart before the horse
To “put the cart before the horse” means to do things in the wrong order. The horse is supposed to be in front, pulling the cart.
How it connects: Substitution has an order too: isolate before you substitute, solve for the first variable before you back-substitute, and check last. The limit of the idiom: the order matters for these steps, but which variable you isolate first does not. Either choice leads to the same solution, with more or less work.
The proof of the pudding is in the eating
This proverb means you can only judge how good something really is by trying it.
How it connects: A solution is only proven by trying it: substitute the pair into both original equations. If both are true, the pair really is the solution. The limit of the proverb: the “eating” is one specific test, substituting into both equations. A pair that merely looks right, or fits only one equation, is not proven.
The team player
Why does the equation y = 5x + 1 make a great teammate? Any time someone needs a substitute for y, it volunteers.
How it connects: y = 5x + 1 says that y and 5x + 1 are the same number, so one can stand in for the other. The limit of the joke: a real substitute player is a different person, but here the stand-in is exactly equal to what it replaces. And the swap goes into the other equation. Putting 5x + 1 back into the equation it came from tells you nothing.
Memory help
SWAP
Solve one equation for a variable. Write that expression, in parentheses, in the other equation. Algebra: solve the new equation, which has one variable. Plug the value back in to find the other variable. Then check both original equations.
Parentheses first
When the expression you substitute has more than one term, wrap it in parentheses: 3x − 2(x − 5), not 3x − 2x − 5. A minus sign in front of the parentheses flips every sign inside.
Use the other equation
You get the expression from one equation, so it goes into the other equation. Putting it back into the equation it came from gives a statement that is always true and teaches you nothing.
Vanish and read
“Variables vanish? Read what’s there: true means endless, false means none anywhere.” A true leftover (like 12 = 12) means infinitely many solutions. A false leftover (like 4 = 7) means no solution.
Box it, then drop it
Draw a box around the expression you got from one equation. In the other equation, circle the variable it equals. Copy the boxed expression in place of the circle, with parentheses around it, and cross out the circle.