ISEE Middle Level

Systems of Equations

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

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Imagine you are a math detective on a top-secret mission! 🕵️‍♂️ You need to figure out the cost of a burger and fries, but the restaurant menu only gives you combo prices. The menu says: 1 Burger + 1 Fries = $10. It also says: 1 Burger - 1 Fries = $4. How much is each item?

Welcome to the world of Systems of Equations! A 'system' is just a fancy way of saying 'two math clues working together to solve a mystery.' If you only have one clue (like $x + y = 10$ ), there are too many possibilities. The burger could be $9 and the fries $1, or the burger could be $6 and the fries $4. But when you get a second clue, there is only one perfect answer that works for both! 🍔🍟

On the ISEE, you will see questions where you have to find the value of two mystery letters, usually $x$ and $y$ . Don't panic! You have two superpowers to solve these: 'Substitution' (swapping things out) and 'Elimination' (adding clues together to make a letter disappear). And remember, the ISEE is a multiple-choice test. If you ever get stuck, you can just plug the answer choices into the equations to see which one makes both clues true. Let's crack the case! 🔍

Key Formula

The Elimination Rule: Stack your equations and add them together to eliminate a variable! If

x + y = 10

and

x - y = 4

, adding them gives

2 x = 14

, so

x = \frac{14}{2} = 7

Practice Questions

4 practice questions for ISEE Middle Level

Q1 Medium

What is the value of

x

in the following system of equations?

x + y = 10

2 x - y = 5

x = 3

x = 4

x = 5

x = 6

Show Solution

1. Given the system of equations:

(1) $x + y = 10$
(2) $2 x - y = 5$
2. Notice that the $y$ terms in both equations have opposite coefficients ( $+ y$ and $- y$ ). We can eliminate $y$ by adding the two equations together:

$(x + y) + (2 x - y) = 10 + 5$
3. Combine the like terms on both sides of the equation:

$x + 2 x + y - y = 15$
$3 x = 15$
4. Divide both sides by 3 to solve for $x$ :

$x = rac{15}{3}$
$x = 5$
To verify, substitute $x = 5$ into the first equation:
$5 + y = 10$
$y = 5$
Then check with the second equation:
$2 (5) - 5 = 10 - 5 = 5$ . The values satisfy both equations.

Answer: C

Q2 Medium

A rectangular garden has a perimeter of 40 feet. The length of the garden is 4 feet more than its width. What is the length of the garden?

A 8 feet

B 10 feet

C 12 feet

D 14 feet

Show Solution

1. Let $L$ represent the length of the garden and $W$ represent the width of the garden.
2. The formula for the perimeter of a rectangle is $P = 2 L + 2 W$ . We are given that the perimeter $P$ is 40 feet, so our first equation is:

(1) $2 L + 2 W = 40$
3. We are also told that the length is 4 feet more than its width. This translates to the second equation:

(2) $L = W + 4$
4. Now we have a system of two equations. We can use the substitution method by substituting the expression for $L$ from equation (2) into equation (1):

$2 (W + 4) + 2 W = 40$
5. Distribute the 2 and simplify the equation:

$2 W + 8 + 2 W = 40$
$4 W + 8 = 40$
6. Subtract 8 from both sides of the equation:

$4 W = 32$
7. Divide by 4 to find the width $W$ :

$W = rac{32}{4}$
$W = 8$ feet
8. The question asks for the length $L$ . Substitute the value of $W$ back into equation (2):

$L = W + 4$
$L = 8 + 4$
$L = 12$ feet
Thus, the length of the garden is 12 feet.

Answer: C

Q3 Medium

What is the value of

y

in the following system of equations?

3 x + 2 y = 13

x - y = 1

y = 1

y = 2

y = 3

y = 4

Show Solution

1. Given the system of equations:

(1) $3 x + 2 y = 13$
(2) $x - y = 1$
2. We want to find the value of $y$ . We can use the elimination method. To eliminate $x$ , multiply equation (2) by 3:

$3 (x - y) = 3 (1)$
(3) $3 x - 3 y = 3$
3. Now subtract equation (3) from equation (1):

$(3 x + 2 y) - (3 x - 3 y) = 13 - 3$
4. Distribute the negative sign and combine like terms:

$3 x + 2 y - 3 x + 3 y = 10$
$5 y = 10$
5. Divide both sides by 5 to solve for $y$ :

$y = rac{10}{5}$
$y = 2$
To verify, substitute $y = 2$ into equation (2) to find $x$ :
$x - 2 = 1$
$x = 3$
Then check with equation (1):
$3 (3) + 2 (2) = 9 + 4 = 13$ . The values satisfy both equations.

Answer: B

Q4 Medium

Sarah bought 3 notebooks and 2 pens for $

11

. Mark bought 1 notebook and 4 pens for $

7

. Assuming all notebooks cost the same and all pens cost the same, what is the cost of one notebook?

A $

2.00

B $

2.50

C $

3.00

D $

3.50

Show Solution

1. Let $n$ represent the cost of one notebook and $p$ represent the cost of one pen.
2. From Sarah's purchase (3 notebooks and 2 pens for $ $11$ ), we can form the equation:

(1) $3 n + 2 p = 11$
3. From Mark's purchase (1 notebook and 4 pens for $ $7$ ), we can form the equation:

(2) $n + 4 p = 7$
4. We want to find the cost of one notebook ( $n$ ). We can use the elimination method. To eliminate $p$ , multiply equation (1) by 2:

$2 (3 n + 2 p) = 2 (11)$
(3) $6 n + 4 p = 22$
5. Now subtract equation (2) from equation (3) to eliminate $p$ :

$(6 n + 4 p) - (n + 4 p) = 22 - 7$
6. Simplify and combine like terms:

$6 n - n + 4 p - 4 p = 15$
$5 n = 15$
7. Divide by 5 to solve for $n$ :

$n = rac{15}{5}$
$n = 3$
So, the cost of one notebook is $ $3.00$ .

Answer: C

Tips & Strategies

Plug in the answer choices! If you're stuck, just try the choices (A, B, C, D) in BOTH equations. If a choice works for both clues, you found the winner! 🏆
Look for opposites. If you see a $+ y$ in one equation and a $- y$ in the other, stack them up and add! They will cancel out like magic.

Common Mistakes

Watch out for finding the wrong variable! If the question asks for $y$ , don't accidentally stop after you find $x$ . Always double-check what the question is asking for before bubbling your answer. 🛑
Don't forget to do the same thing to both sides! If you multiply one side of an equation by 2, you MUST multiply the other side by 2 as well.

Frequently Asked Questions

What if I forget how to do elimination or substitution during the ISEE?

Don't panic! The ISEE is multiple-choice. You can always use the 'guess and check' method by plugging the answer choices into the equations. There is no penalty for guessing, so pick your favorite letter if you're totally stuck!

Will I have to graph these on the test?

Usually not! While graphing is a way to solve systems of equations in school, the ISEE focuses more on algebra and word problems. Stick to substitution, elimination, or plugging in the choices!

What is a Quantitative Comparison question?

It's a special ISEE question type! You get Column A and Column B. You have to decide if A is bigger, B is bigger, they are equal, or if it's impossible to tell. Solve the math system first, then compare your answer to the other column!

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