Lesson 9-3: Write Equations to Represent Relationships Between Two Variables

Warm-Up 1

Which equation represents "y is 4 times x"?

Vocabulary: An equation uses an equal sign to show two quantities are the same. A variable is a letter that stands for a number. "4 times x" means multiply x by 4.

✓Correct! "y is 4 times x" means multiply: y = 4x.

✗Not quite. "4 times x" means multiplication, so the equation is B: y = 4x.

Warm-Up 2

True or false: The equation y = x + 5 means y is always 5 more than x.

Hint: Try substituting a value. If x = 3, then y = 3 + 5 = 8. Is 8 exactly 5 more than 3?

✓Correct! y = x + 5 always adds 5 to whatever x is, so y is always 5 more than x.

✗Actually, y = x + 5 does mean y is always 5 more than x. Try any value of x — you always add 5. The answer is True.

Practice 1

Look at the table below. Write the equation that describes the relationship between x and y.

Strategy: Find the pattern. Look at each pair: when x = 1, y = 7. When x = 2, y = 14. Ask yourself, "What do I multiply x by to get y?" The number you multiply by is the coefficient.

x	1	2	3	4
y	7	14	21	28

y = x

Explain how you found the pattern:

Sentence frame: "I divided each y-value by x and got ___. So the equation is y = ___x."

✓Correct! Each y-value is 7 times the x-value, so y = 7x.

✗Try dividing y by x for each pair: 7 / 1 = 7, 14 / 2 = 7, 21 / 3 = 7. The answer is 7, so y = 7x.

Practice 2

A car travels 60 miles per hour. Which equation shows the distance (d) after h hours?

Key idea: Distance = rate × time. The rate is 60 miles per hour. Time is h. What operation connects rate and time to get distance?

Explain your reasoning:

Sentence frame: "Since the car travels ___ miles each hour, I multiply ___ by ___ to get the total distance."

✓Correct! Distance = rate × time, so d = 60h. After 3 hours the car travels 60 × 3 = 180 miles.

✗Distance equals rate times time. 60 miles per hour × h hours = 60h. The answer is B: d = 60h.

Practice 3

Drag each equation to match its verbal description.

Vocabulary: "Triple" means multiply by 3. "Plus" or "more than" means add. "Half of" means divide by 2. "Subtract" or "less than" means minus.

y = 3x

y = x + 10

y = x / 2

y = x - 4

Triple x

x plus 10

Half of x

4 less than x

✓All matched correctly! Translating words into equations is a key skill for representing relationships.

✗Some matches are incorrect. Remember: "triple" = 3x, "plus 10" = x + 10, "half" = x / 2, "4 less than" = x - 4. Try again!

Practice 4

Look at the table. Write the equation that describes the relationship between x and y.

Strategy: For each row, ask: "What do I add to x to get y?" When x = 2, y = 5. So 2 + ? = 5. The number you add is the constant.

x	2	4	6	8
y	5	7	9	11

y = x +

Explain how you found the constant:

Sentence frame: "I subtracted x from y for each pair: ___ - ___ = ___. The constant is ___, so y = x + ___."

✓Correct! Each y-value is 3 more than x, so y = x + 3.

✗Subtract x from y: 5 - 2 = 3, 7 - 4 = 3, 9 - 6 = 3. The constant is 3, so y = x + 3.

Practice 5

Which table matches the equation y = 2x + 1?

How to check: Substitute each x-value into y = 2x + 1 and see which table's y-values match.
Example: If x = 1, then y = 2(1) + 1 = 3.

Show your substitution work:

Sentence frame: "I substituted each x-value into y = 2x + 1. When x = 1, y = ___. When x = 2, y = ___. Table ___ matches."

✓Correct! y = 2(1)+1 = 3, y = 2(2)+1 = 5, y = 2(3)+1 = 7, y = 2(4)+1 = 9. Table A matches y = 2x + 1.

✗Substitute each x into y = 2x + 1: x=1 gives 3, x=2 gives 5, x=3 gives 7, x=4 gives 9. The answer is A.

Challenge

A babysitter charges $5 plus $8 per hour. Let h = hours and c = total cost.

Step-by-step guide:
1. The babysitter starts with a constant fee of $5 (this never changes).
2. For every hour, add $8 more. The $8 is the coefficient (it multiplies the variable).
3. Write the equation: cost = constant + coefficient × hours.
4. Build a table by substituting h = 1, 2, 3, 4, 5.

Part A: Write the equation (c = ...):

Part B: Complete the table:

Hours (h)	1	2	3	4	5
Cost (c)

Part C: Explain what each part of your equation means:

Sentence frame: "The equation is c = ___. The ___ represents the starting fee. The ___ means the babysitter earns $___ for each hour. So for ___ hours, the total cost is $___."

✓Great work! c = 5 + 8h (or c = 8h + 5). The $5 is the flat fee and the $8 is the hourly rate. The table values are: $13, $21, $29, $37, $45.

✗The equation is c = 5 + 8h (or c = 8h + 5). For each hour, multiply by 8 then add 5: 5+8(1)=13, 5+8(2)=21, 5+8(3)=29, 5+8(4)=37, 5+8(5)=45.

🟠 Level 0 — Extra Support

Warm-Up

Practice

Challenge