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# 5.1 - Solving Systems of Equations

Up until this point, we have been dealing with only one equation at a time. Now, we will workwith more than variable and more than one equation. These are called systems of equations. When answering a system of equations, you need to give the value for each variable.

## Solving Systems of Linear Equations

When we're through covering the two chapters on solving systems of equations, there will be sixways that we can use to solve a system of linear equations

Graphically## Substitution

The method of substitution will work with non-linear as well as linear equations.

- Solve one of the equations for one of the variables.
- Substitute that expression in for the variable in the other equation.
- Solve the equation for the remaining variable
- Back-substitute the value for the variable to find the other variable.
- Check

The process of back-substitution involves taking the value of the variable found in step 3 andsubstituting it back into the expression obtained in step 1 (or the original problem) to find theremaining variable.

It is important that both variables be given when solving a system of equations. A commonmistake students make is to find one variable and stop there. You need to include a value for allthe variables.

It is a good idea to check your answer into the both equations, but is probably sufficient to checkin the equation you didn't isolate a variable in the first step. That is, if you solved for y in thefirst equation in step 1, use the second equation to check the answer.

## Graphical Approach

The graphical approach works well with a graphing calculator, but is inaccurate by hand (didthose points intersect at 1/6 or 1/7?) unless the graph happens to fall exactly on the grid lines.

- Solve each equation for y. This may involve a plus and minus if there is a y
^{2}term. If you'renot graphing with a calculator or computer, you can skip this step. - Graph each equation.
- Find the points of intersection.
- Check!

It is important to check your answers to make sure that you have read the intersection pointcorrectly.

Sometimes the calculator will fail to give an intersection point using the intersect command. You may need to use the trace feature of the calculator to find the intersection point. You mayuse your calculator to check the answer.

Try to convert your answer to fractional form if possible.

The graphical approach can save a lot of time when you're working with a non-linear system ofequations.