➕ Algebra 1 · Systems of Equations
Solving Systems by Substitution flashcards
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All 14 flashcards at a glance
Card 1
What is the idea behind the substitution method?
If one equation says what a variable equals, write that expression in place of the variable in the other equation. Then only one variable is left, and you can solve for it.
Card 2
When is substitution especially convenient?
When a variable is already alone (like y = 2x + 1) or has a coefficient of 1 or −1, so it is easy to isolate. Look in both equations for one.
Card 3
What are the steps of substitution?
Isolate a variable, substitute it (in parentheses) into the other equation, solve, back-substitute to find the other variable, and check in both original equations.
Card 4
Solve y = x + 6 and 2x + y = 3.
(−1, 5). Substitute: 2x + (x + 6) = 3, so 3x + 6 = 3, 3x = −3 and x = −1. Then y = −1 + 6 = 5. Check: 2(−1) + 5 = 3.
Card 5
Substitute x = 4y − 3 into 2x + 5y = 9. Write the new equation (do not solve it).
2(4y − 3) + 5y = 9. The parentheses make the 2 multiply both terms.
Card 6
You found x = 2, and the isolated equation was y = 9 − 4x. What is y?
y = 9 − 4(2) = 1, so the pair is (2, 1). Then check it in both original equations.
Card 7
Solve x − 6y = 2 for x.
Add 6y to both sides: x = 6y + 2.
Card 8
Simplify x − (2y − 5).
x − 2y + 5. The minus sign in front of the parentheses flips every sign inside.
Card 9
After substituting and simplifying you get 6 = 6. What does that mean?
The leftover statement is true, so the system has infinitely many solutions. The two equations describe the same line.
Card 10
After substituting and simplifying you get 2 = 9. What does that mean?
The leftover statement is false, so the system has no solution. The lines are parallel.
Card 11
Why check in both original equations?
A pair is a solution only if it makes both equations true. When the system has exactly one solution, each original equation catches different slips: a sign error while isolating shows up only in the original version of the equation you isolated, and a slip while substituting, solving or back-substituting shows up in at least one of the two. (If both equations describe the same line, a slip can give a pair that fits both, so a passing check does not prove there is only one solution.)
Card 12
You isolate y in the first equation and then substitute it back into the first equation. What happens?
You get a statement that is always true, like 3x − 4 = 3x − 4, and you learn nothing. Substitute into the other equation.
Card 13
Which variable is easiest to isolate in 3x + y = 8 and 5x − 2y = 1?
The y in the first equation, because it has coefficient 1: y = 8 − 3x.
Card 14
In this lesson’s SWAP memory trick, what do the letters stand for?
Solve one equation for a variable, Write that expression (in parentheses) in the other equation, Algebra to solve the one variable left, Plug the value back in to find the other variable. Then check both equations.