Solving Applications of Systems of Equations By Substitution

Solving Applications of Systems of Equations By Substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing lesson for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.

Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?

Example

The sum of two numbers is zero. One number is nine less than the other. Find the numbers.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.	We are looking for two numbers.
Step 3. Name what we are looking for.	Let \(n=\) the first number Let \(m=\) the second number
Step 4. Translate into a system of equations.	The sum of two numbers is zero.
	One number is nine less than the other.
The system is:
Step 5. Solve the system of equations. We will use substitution since the second equation is solved for n.
Substitute m − 9 for n in the first equation.
Solve for m.
Substitute \(m=\frac{9}{2}\) into the second equation and then solve for n.
Step 6. Check the answer in the problem.	Do these numbers make sense in the problem? We will leave this to you!
Step 7. Answer the question.	The numbers are \(\frac{9}{2}\) and \(-\frac{9}{2}.\)

In the the example below, we’ll use the formula for the perimeter of a rectangle, P = 2L + 2W.

Example

The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and the width.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	We are looking for the length and width.
Step 3. Name what we are looking for.	Let \(L=\) the length   \(W=\) the width
Step 4. Translate into a system of equations.	The perimeter of a rectangle is 88.
	2L + 2W = P
	The length is five more than twice the width.
The system is:
Step 5. Solve the system of equations. We will use substitution since the second equation is solved for L. Substitute 2W + 5 for L in the first equation.
Solve for W.
Substitute W = 13 into the second equation and then solve for L.
Step 6. Check the answer in the problem.	Does a rectangle with length 31 and width 13 have perimeter 88? Yes.
Step 7. Answer the equation.	The length is 31 and the width is 13.

For the example below we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.

Example

The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.

Solution

We will draw and label a figure.

Step 1. Read the problem.
Step 2. Identify what you are looking for.	We are looking for the measures of the angles.
Step 3. Name what we are looking for.	Let \(a=\) the measure of the 1st angle \(\phantom{\rule{1.5em}{0ex}}b=\) the measure of the 2nd angle
Step 4. Translate into a system of equations.	The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle.
	The sum of the measures of the angles of a triangle is 180.
The system is:
Step 5. Solve the system of equations. We will use substitution since the first equation is solved for a.
Substitute 3b + 10 for a in the second equation.
Solve for b.
Substitute b = 20 into the first equation and then solve for a.
Step 6. Check the answer in the problem.	We will leave this to you!
Step 7. Answer the question.	The measures of the small angles are 20 and 70.

Example

Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $25,000 plus $15 for each training session. Option B would pay her $10,000 + $40 for each training session. How many training sessions would make the salary options equal?

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	We are looking for the number of training sessions that would make the pay equal.
Step 3. Name what we are looking for.	Let \(s=\) Heather’s salary. \(\phantom{\rule{1.5em}{0ex}}n=\) the number of training sessions
Step 4. Translate into a system of equations.	Option A would pay her $25,000 plus $15 for each training session.
	Option B would pay her $10,000 + $40 for each training session
The system is:
Step 5. Solve the system of equations. We will use substitution.
Substitute 25,000 + 15n for s in the second equation.
Solve for n.
Step 6. Check the answer.	Are 600 training sessions a year reasonable? Are the two options equal when n = 600?
Step 7. Answer the question.	The salary options would be equal for 600 training sessions.

Key Concepts

• Solve a system of equations by substitution
  1. Solve one of the equations for either variable.
  2. Substitute the expression from Step 1 into the other equation.
  3. Solve the resulting equation.
  4. Substitute the solution in Step 3 into one of the original equations to find the other variable.
  5. Write the solution as an ordered pair.
  6. Check that the ordered pair is a solution to both original equations.