SHSAT: Linear equations: write them from words, then solve
Solve equations with variables on both sides, and turn word problems about consecutive numbers, ages, totals and rectangles into equations.
How do you turn a word problem into an equation and solve it without slips?
Short answer
Name the unknown, write the other quantities in terms of it, and balance the equation by doing the same thing to both sides. Then check the answer in the words.
Know first: Combining like terms, Distributing over parentheses
Step 1 of 5
Translate the words, then keep the equation balanced
An equation is a balance: whatever you do to one side, you do to the other. Collect the x terms on one side and the plain numbers on the other, then divide.
In word problems, the hard part comes first: writing the equation. Name the unknown, write every other quantity in terms of it, and only then solve.
| The question says | Write |
|---|---|
| three consecutive integers | n, n + 1, n + 2 |
| two consecutive odd (or even) integers | n, n + 2 |
| a number decreased by 4, then multiplied by 3 | 3(n − 4) |
| 32 tickets, some at $5 and the rest at $8 | x at $8 and 32 − x at $5 |
| in 6 years | add 6 to every person's age |
| a rectangle's perimeter is 54 | 2(length + width) = 54 |
The parentheses matter. 3(n − 4) is not 3n − 4.
Step 2 of 5
Solve, then check both sides
Worked example: Variables on both sides
What value of x satisfies 9x − 14 = 4x + 21?
- A7/5
- B35/13
- C7
- D35
- 1
Collect the x terms
Subtract 4x from both sides: 5x − 14 = 21.
- 2
Collect the numbers
Add 14 to both sides: 5x = 35.
- 3
Divide
x = 7.
- 4
Check
Left: 9(7) − 14 = 49. Right: 4(7) + 21 = 49. Both sides agree.
- 5
See why the other choices are there
7/5 subtracts 14 from 21 instead of adding it. 35/13 adds 9x and 4x. 35 stops before dividing by 5.
Answer
x = 7
Building the equation from a story
A food truck sells 40 meals, some for $6 and the rest for $9. It takes in $303.
Let x = number of $6 meals, so 40 − x are $9 meals.
6x + 9(40 − x) = 303
6x + 360 − 9x = 303, so −3x = −57 and x = 19
$9 meals: 40 − 19 = 21. Check: 6(19) + 9(21) = 114 + 189 = 303
Find the wrong step
A number is increased by 5, and the result is doubled. This equals 3 times the number, minus 7. Find the number.
- 1
Writes 2n + 5 = 3n − 7.
What went wrong
Doubling applies to the whole result, n + 5. The equation needs 2(n + 5).
- 2
Subtracts 2n and adds 7: 12 = n.
- 3
Answers 12.
The fix
2(n + 5) = 3n − 7 gives 2n + 10 = 3n − 7, so n = 17. Check: 17 + 5 = 22, doubled is 44, and 3(17) − 7 = 44.
Step 3 of 5
Write it, solve it, check it
Check yourself · Question 1
Four consecutive integers have a sum of 126. What is the smallest of them?
Answer: A
Write n + (n + 1) + (n + 2) + (n + 3) = 4n + 6 = 126. So 4n = 120 and n = 30. The integers are 30, 31, 32, 33.
Check yourself · Question 2
A father is 4 times as old as his son. In 5 years, the father will be 3 times as old as his son. How old is the son now?
Answer: A
Let s be the son's age. In 5 years, both are 5 years older: 4s + 5 = 3(s + 5). Then 4s + 5 = 3s + 15, so s = 10. Check: now 10 and 40; in 5 years 15 and 45, and 45 = 3 × 15.
Trap answer: 15
Aging only one person
- Why it looks right
- The phrase "in 5 years" sits next to the son in the sentence, so you add 5 there.
- Why it's wrong
- Time passes for both people. The father's age in 5 years is 4s + 5, not 4s.
- The right answer
- 10, from 4s + 5 = 3(s + 5).
Check yourself · Question 3
A rectangle's length is 6 centimeters more than its width. Its perimeter is 52 centimeters. What is its area in square centimeters?
Answer: B
2(w + w + 6) = 52, so 2w + 6 = 26 and w = 10. The length is 16, and the area is 10 × 16 = 160.
Common mistake
Writing 3(n − 4) as 3n − 4.
- Why it's tempting
- The parentheses look like they only group the subtraction, not the multiplication.
- Do this instead
- The 3 multiplies every term inside: 3(n − 4) = 3n − 12. Distribute to each term before you move anything across the equals sign.
Step 4 of 5
What to remember
Remember
- Name the unknown, write every other quantity in terms of it, and keep parentheses where the words group things.
- Collect x terms on one side and numbers on the other, then divide.
- Check your answer in the story itself, and make sure you report the quantity the question asks for.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what did you do first to get all the x terms on one side, and what did both sides equal when you checked?