word problems with linear equations

Question

A garden table and a bench cost $843

 combined. The garden table costs $57

 less than the bench. What is the cost of the bench?

Answer 1

This word problem has relationships between the prices of the objects involved.

Let T denote the price of a garden table and let B denote the price of a bench.

Then the sentence "A garden table and a bench cost $843" can be translated to

the mathematical equation T + B = 843. <1>

The other sentence "The garden table costs $57 less than the bench." can be translated to

the mathematical equation T = B - 57. <2>

Our goal is to answer the question "What is the cost of the bench?" so we need to solve for B.

First, let's use <2> to substitute (B - 57) in the place of T in <1> to obtain

(B - 57) + B = 843

Then solve for B:

2B - 57 = 843

2B = 900

B = 450

So "B = 450" means that the cost of the bench is $450.

As bonus, to solve for T, we can substitute 450 for B in either <1> or <2>. Let's choose <2>:

T = 450 - 57

T = 393

So the cost of the garden table is $393.