# Linear Equation Word Problems

Equations and Word Problems (adding and subtracting)

First, we’ll have to set up our Equation.

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All solutions to Math word problems have to come with equations that are crafted carefully and that describe the constraints that are stated in the problem accurately.

Then we’ll have to solve our Equation.

At all times, you must solve the equation that’s set up (our previous step).

Make sure to answer the question.

Though this may seem obvious, it is a step that’s easily overlooked. For instance, the question may ask you to find Jane’s age, but the solution of your equation gives you the age of Liz, Jane’s sister. Be sure to answer the original question that the problem asks!

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And please be sure that the solution is written in a sentence with appropriate units.

EXAMPLE

Amelie withdraws $$125$$ from her savings account. Because of the withdrawal, the current balance in her account is now $$1,200$$.

What was the original balance in the account before the withdrawal?

$$B\, – 125 = 1200$$ Original equation.

$$B\, – 125 + 125 = 1200 + 125$$

Add $$125$$ to both sides of the equation.

$$B = 1325$$  On the left, adding $$125$$ “undoes” the effect of subtracting $$125$$ and returns $$B$$.

On the right, $$1200+125=1325$$.

Answer the Question. The original balance was $$1,325$$.

EXAMPLE

The perimeter of a triangle is $$114$$ feet. Two of the sides of the triangle measure $$30$$ feet and $$40$$ feet, respectively. The question asks you to determine the measure of the 3rd side of this triangle.

$$114 = x + 30+ 40$$ Our equation.

$$114 = x + 70$$

$$114 – 70 = x + 70 – 70$$

Subtract $$70$$ from both sides.

$$44 = x$$

On the right, subtracting $$70$$ “undoes” the effect of adding $$70$$ and returns to $$x$$. On the left, $$114 – 70 = 44$$.

Answer the Question. The unknown side of our triangle is $$44$$ feet.

Word Problems (Solving Equations by Multiplication and Division)

EXAMPLE

Fifteen times a certain number is $$45$$.

Find the unknown number.

Solution.

Set up the equation.

Solve the Equation.

Write $$15 \cdot x$$ as $$15x$$.

So $$15x = 45$$

Divide both sides of the equation by $$15$$.

$$15x \div 15= 45 \div 15$$

On the left, dividing by $$15$$ “undoes” the effect

of multiplying

$$x = 3$$

Answer the Question. The unknown number is $$3$$.

EXAMPLE

A class of $$23$$ students averaged $$76$$ points on an academic exam.

What is the total number of points that the class accumulated as a whole?

$$T \div 23 = 76$$

Multiply both sides of the equation by $$23$$.

$$23 \cdot T \div 23 = 76 \cdot 23$$

On the left, multiplying by $$23$$ “undoes” the effect of dividing by 23 and returns to $$T$$. On the right, $$76 \cdot 23 = 1748$$.

$$T = 1748$$

Answer the Question. The total points accumulated by the class on the exam is $$1,748$$.

Last Updated on June 13, 2022.