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Math Mission

Linear equations can tell a money story.

A linear equation can show how one amount changes when another amount changes. In y = mx + b, the b is the starting amount, and m tells how much gets added each time.

See an Example
Start money + money earned each time = total money.

What does linear mean?

In 6th grade, a linear equation often looks like y = mx + b. The variable is only to the first power, so the pattern grows in a steady, predictable way.

  1. x: The input. It tells how many times something happens, like chores completed.
  2. m: The rate. It tells how much is added for each 1 more x. This is called the slope.
  3. b: The starting amount. It is what you already have before x even begins.

Example problem

Milo is saving for a glow-in-the-dark skateboard. He gets $5 every week for helping at home, and he earns 75 cents for each pet chore he finishes. If he does 4 chores, how much money does he make that week?

y = 0.75(4) + 5 = ?

First write the story as an equation: y = 0.75x + 5. The 5 is the starting money, and 0.75 is the money earned per chore.

Step 1 Write the rule: y = 0.75x + 5. This means total money equals chore money plus the starting $5.
Step 2 Plug in x = 4 because Milo did 4 chores.
Step 3 Multiply: 0.75 x 4 = 3. Milo earned $3 from chores.
Step 4 Add the starting $5: 3 + 5 = 8. Milo makes $8 that week.

Solving recipe

  1. Name the parts. In y = mx + b, m is the rate and b is the starting amount.
  2. Plug in x. Replace x with the number from the story problem.
  3. Multiply first. Find how much the variable part adds to the total.
  4. Add the start amount. Finish with b to get the total y.

Another quick one

y = 1.50x + 2

Imagine a game arcade card with $2 already loaded, and then you add $1.50 for each token pack. If you buy 3 packs, then y = 1.50(3) + 2 = 6.50.

Mini check

The total keeps growing by the same amount each time, so it is still linear.

Solving for x when y is known

Sometimes you already know the total money, and you need to figure out how many chores were done. This means you solve for x by working backward.

14 = 0.75x + 5 → x = ?

Milo has $14 total. His rule is still y = 0.75x + 5. To find x, first remove the starting $5, then divide by 0.75.

Step 1 Start with the equation 14 = 0.75x + 5.
Step 2 Subtract 5 from both sides: 14 - 5 = 0.75x, so 9 = 0.75x.
Step 3 Divide both sides by 0.75: 9 ÷ 0.75 = x.
Step 4 x = 12. Milo did 12 chores to make $14 total.

Backward recipe

  1. Write the total as y. Put the known total in place of y.
  2. Subtract b first. Remove the starting amount because it was there before any x happened.
  3. Divide by m. This undoes the amount added for each x.
  4. Check the answer. Plug your x back into the original equation to see if it makes the correct total y.

Why subtract first?

In y = mx + b, the b is extra money already sitting there at the start. If you want only the part made by chores, you have to peel off the starting money first.

Think of it like this

If someone hands you a jar with $14 in it, and you know $5 was there before the chores, then only $9 came from chores.

Try one yourself

Zoe runs a weekend lemonade stand. She starts with $4 in her jar, and she earns $1.25 for each lemonade she sells. If she sells 3 lemonades, what is y?

Hint Press the hint button if you want a nudge.
Remember Multiply the amount per lemonade by 3 first, then add the starting $4.
Check Your answer should be more than $4, because Zoe started with money before selling anything.

Now solve for x

Theo’s pet-walking rule is y = 2x + 6. He has $16 total. How many dog walks did he do?

Hint Press the hint button if you want a nudge.
Remember Start by subtracting the +6, because that is the starting amount.
Check Your answer should make 2x + 6 equal 16.

Start amount, steady rate, total money. That is the heart of y = mx + b.