SSAT Middle Level

Systems of Equations

Q: How do I know whether to use substitution or elimination?

If one equation already has a letter all by itself (like `y = 2x + 1`), substitution is usually the fastest. If both equations have the letters mixed together (like `3x + 2y = 10`), try stacking them for elimination!

Solving for two or more unknowns simultaneously using substitution, elimination, or graphing

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Imagine you are at a carnival snack stand. You buy 2 giant pretzels and 1 soda for $8. Your best friend buys 1 giant pretzel and 2 sodas for $7. 🥨🥤 How much does a single pretzel cost? How about one soda? You don't know the individual prices, but you have two great clues!

Welcome to Systems of Equations! It sounds like a super complicated robot term, but it's really just a fancy math phrase for solving two puzzles at the exact same time. On the SSAT, you will get to be a math detective. Instead of pretzels and sodas, you usually get clues with the letters $x$ and $y$ . A system of equations happens when you have two unknown numbers and two math sentences (equations) that give you hints about them.

To crack the case, you have two superpower moves. Move number one is called Substitution. This is when you know exactly what one letter equals, so you swap it into the other equation like a secret agent in disguise! 🕵️‍♂️ Move number two is Elimination. This is when you stack the two equations on top of each other and add or subtract them so that one of the letters completely vanishes! Poof! 🪄 Once you find the hidden number for the first letter, you just plug it back in to find the second one. You've got this!

Key Formula

To solve by substitution: If

y = 2 x + 1

and

x + y = 10

, replace

y

in the second equation to get

x + (2 x + 1) = 10

. Solve for

x

, then plug it back in to find

y

Practice Questions

4 practice questions for SSAT Middle Level

Q1 Medium

What is the value of

x

in the system of equations below?

x + y = 15

x - y = 3

A 4

B 6

C 9

D 12

E 18

Show Solution

We are given the system of equations:
1. $x + y = 15$
2. $x - y = 3$

To solve for $x$ , we can use the elimination method. Notice that the $y$ terms have opposite signs.
Step 1: Add Equation 1 and Equation 2 together:

$(x + y) + (x - y) = 15 + 3$
$x + y + x - y = 18$
Step 2: Combine like terms. The $y$ terms cancel out:

$2 x = 18$
Step 3: Divide both sides by 2 to solve for $x$ :

$x = \frac{18}{2}$
$x = 9$
Thus, the value of $x$ is 9.

Answer: C

Q2 Medium

Mia is 4 years older than her brother, Leo. The sum of their ages is 20. How old is Leo?

A 6

B 8

C 10

D 12

E 14

Show Solution

Let $M$ represent Mia's age and $L$ represent Leo's age.
Step 1: Translate the first sentence into an equation. "Mia is 4 years older than her brother, Leo" means:

$M = L + 4$ (Equation 1)
Step 2: Translate the second sentence into an equation. "The sum of their ages is 20" means:

$M + L = 20$ (Equation 2)
Step 3: Use the substitution method. Substitute Equation 1 into Equation 2:

$(L + 4) + L = 20$
Step 4: Combine like terms:

$2 L + 4 = 20$
Step 5: Subtract 4 from both sides:

$2 L = 20 - 4$
$2 L = 16$
Step 6: Divide by 2 to solve for $L$ :

$L = \frac{16}{2}$
$L = 8$
Leo is 8 years old.

Answer: B

Q3 Medium

Given the system of equations:

y = 2 x - 1

3 x + y = 14

What is the value of

x + y

A 3

B 5

C 8

D 11

E 13

Show Solution

We are given the system of equations:
1. $y = 2 x - 1$
2. $3 x + y = 14$
Step 1: Use the substitution method. Since Equation 1 already defines $y$ in terms of $x$ , substitute $(2 x - 1)$ for $y$ in Equation 2:

$3 x + (2 x - 1) = 14$
Step 2: Combine like terms:

$5 x - 1 = 14$
Step 3: Add 1 to both sides:

$5 x = 14 + 1$
$5 x = 15$
Step 4: Divide by 5 to solve for $x$ :

$x = \frac{15}{5}$
$x = 3$
Step 5: Substitute the value of $x$ (which is 3) back into Equation 1 to find $y$ :

$y = 2 (3) - 1$
$y = 6 - 1$
$y = 5$
Step 6: The question asks for the value of $x + y$ :

$x + y = 3 + 5$
$x + y = 8$
Thus, the value of $x + y$ is 8.

Answer: C

Q4 Medium

A local cafe sells coffee and tea. On Monday, they sold a total of 50 drinks. Each coffee costs $2, and each tea costs $3. If the total revenue from these 50 drinks was $120, how many coffees were sold?

A 10

B 20

C 30

D 40

E 45

Show Solution

Let $c$ represent the number of coffees sold and $t$ represent the number of teas sold.
Step 1: Formulate equations based on the given information.

"On Monday, they sold a total of 50 drinks" leads to:
$c + t = 50$ (Equation 1)
"Each coffee costs $2, and each tea costs $3. ... total revenue ... was $120" leads to:
$2 c + 3 t = 120$ (Equation 2)
Step 2: From Equation 1, express $t$ in terms of $c$ :

$t = 50 - c$
Step 3: Substitute this expression for $t$ into Equation 2:

$2 c + 3 (50 - c) = 120$
Step 4: Distribute the 3:

$2 c + 150 - 3 c = 120$
Step 5: Combine like terms:

$- c + 150 = 120$
Step 6: Subtract 150 from both sides:

$- c = 120 - 150$
$- c = - 30$
Step 7: Multiply both sides by -1 to solve for $c$ :

$c = 30$
Therefore, 30 coffees were sold.

Answer: C

Tips & Strategies

Use the answers! If you get stuck on the SSAT, just take the multiple-choice answers and plug them into the equations. The correct answer must make BOTH equations true.
Stack your blocks neatly. Before you add or subtract equations for elimination, make sure your $x$ s, $y$ s, and equal signs are lined up perfectly.

Common Mistakes

Watch out for the half-finished trap! You might do all the hard work to find $x$ , circle it, and move on. But wait... the question might have asked for $y$ ! Always re-read what the question is asking for.
Don't forget to distribute the negative sign! If you subtract an entire equation during elimination, make sure you subtract every single piece of it, not just the first letter.

Frequently Asked Questions

How do I know whether to use substitution or elimination?

If one equation already has a letter all by itself (like $y = 2 x + 1$ ), substitution is usually the fastest. If both equations have the letters mixed together (like $3 x + 2 y = 10$ ), try stacking them for elimination!

Will I have to graph these on the SSAT?

Usually, no! The SSAT focuses on solving these with algebra or through word problems. You won't need to draw a graph yourself, but it's helpful to remember that the solution ( $x$ and $y$ ) is the exact point where the two lines cross on a map.

What if I completely forget how to do the algebra during the test?

Don't panic! Use the magic of SSAT multiple choice. Take the answer choices and plug them in for $x$ and $y$ . If they make both math sentences true, you found the winner without doing any heavy algebra!

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