The elimination method is an algebraic method for solving systems of equations where like terms are aligned.

In this system, the equations are aligned:\begin{aligned} 3x+4y&=12\\\ x+y&=11\end{aligned} and in this system, the equations are not aligned: \begin{aligned} 3x&=12-4y\\ x&+y=11\end{aligned}

Begin by graphing the system of equations and identifying the solution.\begin{aligned} 3x+2y&=10\\x+2y&=8\end{aligned}

Next, perform each operation using the original equations and consider the result. What do you notice? Record your observations.

- Divide the second equation by 2 and regraph the system. What do you notice?
- Multiply the first equation by 2 and regraph the system. What do you notice?
- Add the equations together and regraph the system. What do you notice?
- Multiply the first equation by -1 and add the equations together. Graph the new equation with the system. What do you notice?
- Multiply the second equation by 3 and add the equations together. Graph the new equation with the system. What do you notice?
- Multiply the second equation by -3 and add the equations together. Graph the new equation with the system. What do you notice?

What happens to the solution to a system of linear equations if one equation is multiplied by a number and then added to the other equation?

The goal of the elimination method is to combine the two equations in the system of equations until one variable is eliminated from the problem. When we replace an equation in a system of equations with the sum of that and the multiple of the other, we get a new system with the same solution. This becomes the basis of the elimination method.

We can also determine the number of solution from a system of equations without solving the system. If one equation in the system is a multiple of the other, the system has infinite solutions:

2y - 3 = 4

6y - 9 = 12

This system has infinite solutions because they are different representations of the same line.

If the coefficients of the x and y terms are multiples but the constant is not a multiple, the system has no solutions:

2y - 3 = 4

6y - 9 = 5

This system has no solutions because they represent parallel lines and will never cross.

Lastly, if the x and y terms are not multiples then the lines intersect and have one solution:

2y - 3 = 4

6y - 6 = 5

This system represents lines that intersect exactly once, so there will be a single solution.

Consider the system of equations: \begin{aligned} 4x+y&=11 \\x-2y&=5\end{aligned}

a

Equation 1 is multiplied by 2 to produce the system: \begin{aligned} 8x+2y&=22 \\x-2y&=5\end{aligned} Determine whether or not the two systems are equivalent. Then decide whether or not the two systems result in the same solution.

Worked Solution

b

Equation 2 is replaced with the sum of Equation 2 and 3 times Equation 1 to produce the system: \begin{aligned} 4x+y &=11\\13x+y&=38 \end{aligned} Determine whether or not the two systems are equivalent. Then decide whether or not the two systems result in the same solution.

Worked Solution

Solve each system of equations using the elimination method.

a

\begin{aligned} 9x+y &=62 \\ 5x+y&=38 \end{aligned}

Worked Solution

b

\begin{aligned} 2x+3y &= 19 \\ 4x-y&=10\end{aligned}

Worked Solution

c

\begin{aligned} 4x+8y &= 48 \\ 3x-6y&=18\end{aligned}

Worked Solution

When comparing test results, Verna noticed that the sum of her Chemistry and English test scores was 128 and that their difference was 16. She scored higher on her Chemistry test.

a

Write a system of equations for this scenario, where x represents Verna's Chemistry test score, and y represents her English test score.

Worked Solution

b

Solve the system of equations to find her test scores.

Worked Solution

c

Does the solution make sense in terms of the context? Explain your answer.

Worked Solution

Determine the number of solutions to each system without solving.

a

\begin{aligned} 4x+y=11 \\ x-2y=5 \end{aligned}

Worked Solution

b

\begin{aligned} 2x+3y=-1 \\ 4x+6y=-2 \end{aligned}

Worked Solution

c

\begin{aligned} 2x-4y=0 \\ x-2y=5 \end{aligned}

Worked Solution

Idea summary

The goal of the elimination method is to combine the equations in a system until one variable is eliminated. We can eliminate a variable in a system of equations by multiplying one (or both) equations so that the coefficients of one variable are equal and opposite.