how to solve word problems with 3 variables

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Solving a word problem with 3 unknowns using a linear equation

Amanda, Henry, and Scott have a total of $89 in their wallets. Amanda has $6 less than Scott. Henry has 3 times what Scott has. How much does each have? Solution Let x be the amount of money Amanda has Let y be the amount of money Henry has Let z be the amount of money Scott has Amanda, Henry, and Scott have a total of $89 in their wallets. The above statement gives the following equation x + y + z = 89 Amanda has $6 less than Scott The above statement gives the following equation x = z - 6 Henry has 3 times what Scott has. The above statement gives the following equation y = 3z We get the following 3 equations x + y + z = 89 equation 1 x = z - 6 equation 2 y = 3z equation 3 Replace x = z - 6 and y = 3z in equation 1 z - 6 + 3z + z = 89 5z - 6 = 89 5z - 6 + 6 = 89 + 6 5z = 95 Divide both sides by 5 5z/ 5 = 95 / 5 z = 19 Scott has 19 dollars y = 3z = 3 × 19 = 57 Henry has 57 dollars z - 6 = x 19 - 6 = x 13 = x Amanda has 13 dollars

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Systems of Linear Equations

Solve Systems of Equations with Three Variables

Learning objectives.

By the end of this section, you will be able to:

Before you get started, take this readiness quiz.

Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables

In this section, we will extend our work of solving a system of linear equations. So far we have worked with systems of equations with two equations and two variables. Now we will work with systems of three equations with three variables. But first let’s review what we already know about solving equations and systems involving up to two variables.

Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions

We know when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown.

Figure shows three graphs. In the first one, two lines intersect. Intersecting lines have one point in common. There is one solution to this system. The graph is labeled Consistent Independent. In the second graph, two lines are parallel. Parallel lines have no points in common. There is no solution to this system. The graph is labeled inconsistent. In the third graph, there is just one line. Both equations give the same line. Because we have just one line, there are infinitely many solutions. It is labeled consistent dependent.

A linear equation with three variables, where a, b, c, and d are real numbers and a, b , and c are not all 0, is of the form

All the points that are solutions to one equation form a plane in three-dimensional space. And, by finding what the planes have in common, we’ll find the solution to the system.

When we solve a system of three linear equations represented by a graph of three planes in space, there are three possible cases.

Eight figures are shown. The first one shows three intersecting planes with one point in common. It is labeled Consistent system and Independent equations. The second figure has three parallel planes with no points in common. It is labeled Inconsistent system. In the third figure two planes are coincident and parallel to the third plane. The planes have no points in common. In the fourth figure, two planes are parallel and each intersects the third plane. The planes have no points in common. In the fifth figure, each plane intersects the other two, but all three share no points. The planes have no points in common. In the sixth figure, three planes intersect in one line. There is just one line, so there are infinitely many solutions. In the seventh figure, two planes are coincident and intersect the third plane in a line. There is just one line, so there are infinitely many solutions. In the last figure, three planes are coincident. There is just one plane, so there are infinitely many solutions.

To determine if an ordered triple is a solution to a system of three equations, we substitute the values of the variables into each equation. If the ordered triple makes all three equations true, it is a solution to the system.

$\left\{\begin{array}{c}x-y+z=2\hfill \\ 2x-y-z=-6\hfill \\ 2x+2y+z=-3\hfill \end{array}.$

Solve a System of Linear Equations with Three Variables

To solve a system of linear equations with three variables, we basically use the same techniques we used with systems that had two variables. We start with two pairs of equations and in each pair we eliminate the same variable. This will then give us a system of equations with only two variables and then we know how to solve that system!

Next, we use the values of the two variables we just found to go back to the original equation and find the third variable. We write our answer as an ordered triple and then check our results.

$\left\{\begin{array}{c}x-2y+z=3\hfill \\ 2x+y+z=4\hfill \\ 3x+4y+3z=-1\hfill \end{array}.$

The steps are summarized here.

If any coefficients are fractions, clear them.

Decide which variable you will eliminate.

Work with a pair of equations to eliminate the chosen variable.

Multiply one or both equations so that the coefficients of that variable are opposites.

$\left\{\begin{array}{c}3x-4z=0\hfill \\ 3y+2z=-3\hfill \\ 2x+3y=-5\hfill \end{array}.$

We check that the solution makes all three equations true.

$\begin{array}{ccccccc}\begin{array}{ccc}\hfill 3x-4z& =\hfill & 0\left(1\right)\hfill \\ \hfill 3\left(-4\right)-4\left(-3\right)& \stackrel{?}{=}\hfill & 0\hfill \\ \hfill 0& =\hfill & 0✓\hfill \end{array}\hfill & & & \begin{array}{ccc}\hfill 3y+2z& =\hfill & -3\left(2\right)\hfill \\ \hfill 3\left(1\right)+2\left(-3\right)& \stackrel{?}{=}\hfill & -3\hfill \\ \hfill -3& =\hfill & -3✓\hfill \end{array}\hfill & & & \begin{array}{}\\ \\ \begin{array}{ccc}\hfill 2x+3y& =\hfill & -5\left(3\right)\hfill \\ \hfill 2\left(-4\right)+3\left(1\right)& \stackrel{?}{=}\hfill & -5\hfill \\ \hfill -5& =\hfill & -5✓\hfill \end{array}\hfill \\ \text{The solution is}\phantom{\rule{0.2em}{0ex}}\left(-4,1,-3\right).\hfill \end{array}\hfill \end{array}$

When we solve a system and end up with no variables and a false statement, we know there are no solutions and that the system is inconsistent. The next example shows a system of equations that is inconsistent.

$\left\{\begin{array}{c}x+2y-3z=-1\hfill \\ x-3y+z=1\hfill \\ 2x-y-2z=2\hfill \end{array}.$

Use equation (1) and (2) to eliminate z .

The equations are x plus 2y minus 3z equals minus 1, x minus 3y plus z equals 1 and 2x minus y minus 2z equals 2.

Use (4) and (5) to eliminate a variable.

Equations 4 and 5 both have 2 variables. Multiply equation 5 by minus 1 and add it to equation 4. We get 0 equal to minus 2, which is false.

There is no solution.

We are left with a false statement and this tells us the system is inconsistent and has no solution.

$\left\{\begin{array}{c}x+2y+6z=5\hfill \\ -x+y-2z=3\hfill \\ x-4y-2z=1\hfill \end{array}.$

no solution

$\left\{\begin{array}{c}2x-2y+3z=6\hfill \\ 4x-3y+2z=0\hfill \\ -2x+3y-7z=1\hfill \end{array}.$

When we solve a system and end up with no variables but a true statement, we know there are infinitely many solutions. The system is consistent with dependent equations. Our solution will show how two of the variables depend on the third.

$\left\{\begin{array}{c}x+2y-z=1\hfill \\ 2x+7y+4z=11\hfill \\ x+3y+z=4\hfill \end{array}.$

Use equation (1) and (3) to eliminate x .

The equations are x plus 2y minus z equals 1, 2x plus 7y plus 4z equals 11 and x plus 3y plus z equals 4. Multiply equation 1 with minus 1 and add it to equation 3. We get equation 4, y plus 2z equals 3.

Use equation (1) and (2) to eliminate x again.

Multiply equation 1 with minus 2 and add it to equation 2. We get equation 5, 3y plus 6z equals 9.

Solve Applications using Systems of Linear Equations with Three Variables

Applications that are modeled by a systems of equations can be solved using the same techniques we used to solve the systems. Many of the application are just extensions to three variables of the types we have solved earlier.

The community college theater department sold three kinds of tickets to its latest play production. The adult tickets sold for ?15, the student tickets for ?10 and the child tickets for ?8. The theater department was thrilled to have sold 250 tickets and brought in ?2,825 in one night. The number of student tickets sold is twice the number of adult tickets sold. How many of each type did the department sell?

The community college fine arts department sold three kinds of tickets to its latest dance presentation. The adult tickets sold for ?20, the student tickets for ?12 and the child tickets for ?10.The fine arts department was thrilled to have sold 350 tickets and brought in ?4,650 in one night. The number of child tickets sold is the same as the number of adult tickets sold. How many of each type did the department sell?

The fine arts department sold 75 adult tickets, 200 student tickets, and 75 child tickets.

The community college soccer team sold three kinds of tickets to its latest game. The adult tickets sold for ?10, the student tickets for ?8 and the child tickets for ?5. The soccer team was thrilled to have sold 600 tickets and brought in ?4,900 for one game. The number of adult tickets is twice the number of child tickets. How many of each type did the soccer team sell?

The soccer team sold 200 adult tickets, 300 student tickets, and 100 child tickets.

Access this online resource for additional instruction and practice with solving a linear system in three variables with no or infinite solutions.

Key Concepts

Practice makes perfect.

In the following exercises, determine whether the ordered triple is a solution to the system.

$\left\{\begin{array}{c}2x-6y+z=3\hfill \\ 3x-4y-3z=2\hfill \\ 2x+3y-2z=3\hfill \end{array}$

In the following exercises, solve the system of equations.

$\left\{\begin{array}{c}5x+2y+z=5\hfill \\ -3x-y+2z=6\hfill \\ 2x+3y-3z=5\hfill \end{array}$

In the following exercises, solve the given problem.

The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is twice the measure of the first angle. The third angle is twelve more than the second. Find the measures of the three angles.

The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is three times the measure of the first angle. The third angle is fifteen more than the second. Find the measures of the three angles.

After watching a major musical production at the theater, the patrons can purchase souvenirs. If a family purchases 4 t-shirts, the video, and 1 stuffed animal, their total is ?135.

A couple buys 2 t-shirts, the video, and 3 stuffed animals for their nieces and spends ?115. Another couple buys 2 t-shirts, the video, and 1 stuffed animal and their total is ?85. What is the cost of each item?

The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of ?65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of ?140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 cans of popcorn for a total sales of ?250. What is the cost of each item?

?20, ?5, ?10

Writing Exercises

In your own words explain the steps to solve a system of linear equations with three variables by elimination.

How can you tell when a system of three linear equations with three variables has no solution? Infinitely many solutions?

Answers will vary.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 columns, 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first row contains the following statements: determine whether an ordered triple is a solution of a system of three linear equations with three variables, solve a system of linear equations with three variables, solve applications using systems of linear equations with three variables. The remaining columns are blank.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Intermediate Algebra by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Word problems relating 3 variable systems of equations

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Find the equation of the parabola, y = a x 2 + b x + c y = ax^{2} + bx + c y = a x 2 + b x + c , that passes through (-2, -10), (1, -4), (3, 30).

The sum of the digits of a three-digit number is 16. The units digit is 2 more than the sum of the other two digits, and the tens digit is 5 more than the hundreds digit. What is the number?

The measure of the largest angle of a triangle is 30° less than the sum of the measures of the other two angles and 9° less than 2 times the measure of the smallest angle. Find the measures of the three angles of the triangle.

Thomas has $6000 invested among a checking account paying 2% annual interest, a savings account paying 5% annual interest, and a bond paying 7% annual interest. He earns a total of $355 in annual interest and he has $2300 less invested in his savings account than in her bond. How much has he invested in each account?

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