Before you read

Answer first — the lesson below explains it.

Shop A rents bikes for a $$3$ fee plus $$1$ per hour, and Shop B charges a $$1$ fee plus $$2$ per hour. For a rental of $x$ hours, the total cost $y$ , in dollars, is $y = x + 3$ at Shop A and $y = 2 x + 1$ at Shop B. Which ordered pair $(x, y)$ is the solution to this system?

A
$(2, 5)$
B
$(1, 3)$
C
$(3, 7)$
D
$(- 2, 1)$

Check your answer

Answer:

Before you read

Answer first — the lesson below explains it.

A customer buys $x$ bags of bird seed at $$3$ per bag and $x$ bags of fish food at $$4$ per bag. The fish food costs $y$ dollars in all, so $y = 4 x$ , and the customer spends $$21$ in total, so $3 x + y = 21$ . Which ordered pair $(x, y)$ is the solution to this system?

A
$(4, 16)$
B
$(21, 84)$
C
$(3, 12)$
D
$(7, 28)$

Check your answer

Answer:

Theory

What Is a System of Equations?

A system of equations is a set of two (or more) equations with the same variables. The solution is the values of the variables that make both equations true at the same time.

Graphically, the solution is the point where the two lines intersect.

System solution at intersection (2, 5)

$\begin{cases} y = 2x + 1 \\ y = -x + 7 \end{cases}$

The solution is the point $(x, y)$ that satisfies both equations simultaneously.

$2x + 1 = -x + 7$ substitute y = 2x + 1
$3x = 6$
$x = 2$ solve
$y = 2(2) + 1 = 5$

Substitution method

Quick check

You just read it. Can you apply it?

Harper's age is $1$ year less than twice Logan's age ( $x = 2 y - 1$ ). Together, a calculation gives $3 x + 4 y = 17$ . What is Harper's age ( $x$ )?

A
$5$
B
$2$
C
$1$
D
$3$

Check your answer

Answer:

Substitute $x = 2 y - 1$ into equation 2:
$3 (2 y - 1) + 4 y = 17$
$6 y - 3 + 4 y = 17$
$10 y = 20$ $\to$ $y = 2$ .
$x = 2 (2) - 1 = 3$ .

Theory

The Substitution Method

When to use: when one equation already has a variable isolated (e.g., $y = ...$ ) or can easily be solved for one variable.

Steps:
1. Isolate one variable in one equation (if not already done)
2. Substitute that expression into the other equation
3. Solve the resulting one-variable equation
4. Back-substitute to find the other variable
5. Check your answer in both equations

Example 1

Solve the system:
$\begin{cases} y = 3x - 1 \\ 2x + y = 9 \end{cases}$

Step 1: $y$ is already isolated in equation 1.

Step 2: Substitute $y = 3x - 1$ into equation 2:

$2x + (3x - 1) = 9$

Step 3: Solve: $5x - 1 = 9$ $\to$ $5x = 10$ $\to$ $x = 2$

Step 4: Back-substitute: $y = 3(2) - 1 = 5$

Step 5: Check in equation 2: $2(2) + 5 = 9$ ✓

(x, y) = (2, 5)

Example 2

Solve the system:
$\begin{cases} x + 2y = 8 \\ 3x - y = 3 \end{cases}$

Step 1: Isolate $x$ from equation 1: $x = 8 - 2y$

Step 2: Substitute into equation 2:

$3(8 - 2y) - y = 3$

Step 3: $24 - 6y - y = 3$ $\to$ $24 - 7y = 3$ $\to$ $-7y = -21$ $\to$ $y = 3$

Step 4: $x = 8 - 2(3) = 2$

Step 5: Check: $3(2) - 3 = 3$ ✓

(x, y) = (2, 3)

Quick check

You just read it. Can you apply it?

The sum of two test scores is $10$ ( $x + y = 10$ ). One score is $2$ less than $3$ times the other ( $y = 3 x - 2$ ). What is $y$ ?

A
$4$
B
$3$
C
$7$
D
$8$

Check your answer

Answer:

Substitute $y = 3 x - 2$ into $x + y = 10$ :
$x + 3 x - 2 = 10$ $\to$ $4 x = 12$ $\to$ $x = 3$ .
$y = 3 (3) - 2 = 7$ .

Common Mistakes

Common Mistakes to Avoid

Substituting back into the same equation

Isolating $y$ from equation 1, then substituting back into equation 1

Always substitute into the OTHER equation.

Forgetting to find the second variable

Finding $x = 2$ and stopping there

You need BOTH

x

and

y

. After finding one, plug it back in to find the other.

Distributing errors during substitution

$3(8 - 2y) = 24 - 2y$ $\leftarrow$ forgot to distribute $3$ to $-2y$

Multiply the coefficient by EVERY term inside the parentheses.

Quick check

You just read it. Can you apply it?

${2 x + y = 11 x - y = 1$
What is the value of $x + y$ ?

A
$7$
B
$8$
C
$9$
D
$11$

Check your answer

Answer:

From equation 2: $x = y + 1$ .
Substitute: $2 (y + 1) + y = 11$ $\to$ $3 y + 2 = 11$ $\to$ $y = 3$ .
$x = 3 + 1 = 4$ .
$x + y = 4 + 3 = 7$ .

Strategy

SAT Strategy Tip

On the SAT, substitution is fastest when one equation is already solved for a variable (like $y = ...$ ). If neither equation is solved, check if elimination might be quicker. Also, the SAT sometimes asks for an expression like $x + y$ or $2x - y$ rather than individual values — you might be able to find the expression directly without finding $x$ and $y$ separately.

After reading

Same question as before. What do you say now?

A
$(2, 5)$
B
$(1, 3)$
C
$(3, 7)$
D
$(- 2, 1)$

Check your answer

Answer:

Both equal $y$ , so: $x + 3 = 2 x + 1$ $\to$ $2 = x$ .
$y = 2 + 3 = 5$ .
Solution: $(2, 5)$ : a $2$ -hour rental costs $$5$ at either shop.
Check: $5 = 2 (2) + 1 = 5$ ✓

After reading

Same question as before. What do you say now?

A
$(4, 16)$
B
$(21, 84)$
C
$(3, 12)$
D
$(7, 28)$

Check your answer

Answer:

Substitute $y = 4 x$ into equation 2: $3 x + 4 x = 21$ $\to$ $7 x = 21$ $\to$ $x = 3$ .
$y = 4 (3) = 12$ . Solution: $(3, 12)$ , so the customer bought $3$ bags of each and spent $$12$ on fish food.
Check: $3 (3) + 12 = 21$ ✓

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