Solving a system of simultaneous equations seems like a very daunting task at first. With more than one unknown quantity to find the value for, and apparently very little way of disentangling one variable from another, it can be a headache for people new to algebra. However, there are three different methods for finding the solution to the equation, with two depending more on algebra and being a bit more reliable, and the other turning the system into a series of lines on a graph.

## Solving a System of Equations by Substitution

## Put One Variable in Terms of the Other

## Substitute the New Expression Into the Other Equation

## Re-arrange and Solve for the First Variable

## Use Your Result to Find the Second Variable

**Check Your Answers**It’s good practice to

*always*check that your answers make sense and work with the original equations. In this example,*x*–*y*= 5, and the result gives 3 – (−2) = 5, or 3 + 2 = 5, which is correct. The second equation states: 3_x_ + 2_y_ = 5, and the result gives 3 × 3 + 2 × (−2) = 9 – 4 = 5, which is again correct. If something doesn’t match up at this stage, you have made a mistake in your algebra.

Solve a system of simultaneous equations by substitution by first expressing one variable in terms of the other. Using these equations as an example:

*x** *– *y* = 5

3_x_ + 2_y_ = 5

Re-arrange the simplest equation to work with and use this to insert into the second. In this case, adding *y* to both sides of the first equation gives:

*x** *= *y* + 5

Use the expression for *x* in the second equation to produce an equation with a single variable. In the example, this makes the second equation:

3 × (*y* + 5) + 2_y_ = 5

3_y_ + 15 + 2_y_ = 5

Collect the like terms to get:

5_y_ + 15 = 5

Re-arrange and solve for *y*, starting by subtracting 15 from both sides:

5_y_ = 5 – 15 = −10

Dividing both sides by 5 gives:

*y** *= −10 ÷ 5 = −2

So *y* = −2.

Insert this result into either equation to solve for the remaining variable. At the end of step 1, you found that:

*x** *= *y* + 5

Use the value you found for *y* to get:

*x** *= −2 + 5 = 3

So *x* = 3 and *y* = −2.

#### Tips

## Solving a System of Equations by Elimination

## Choose a Variable to Eliminate and Adjust the Equations as Needed

## Eliminate One Variable and Solve for the Other

## Use Your Result to Find the Second Variable

Look at your equations to find a variable to remove:

*x** *– *y* = 5

3_x_ + 2_y_ = 5

In the example, you can see that one equation has -* y* and the other has +2_y_. If you add twice the first equation to the second one, the

*y*terms would cancel out and

*y*would be eliminated. In other cases (e.g., if you wanted to eliminate

*x*), you can also subtract a multiple of one equation from the other.

Multiply the first equation by two to prepare it for the elimination method:

2 × (*x* – *y*) = 2 × 5

So

2_x_ – 2_y_ = 10

Eliminate your chosen variable by adding or subtracting one equation from the other. In the example, add the new version of the first equation to the second equation to get:

3_x_ + 2_y_ + (2_x_ – 2_y_) = 5 + 10

3_x_ + 2_x_ + 2_y_ – 2_y_ = 15

So this means:

5_x_ = 15

Solve for the remaining variable. In the example, divide both sides by 5 to get:

* x* = 15 ÷ 5 = 3

As before.

Like in the previous approach, when you have one variable, you can insert this into either expression and re-arrange to find the second. Using the second equation:

3_x_ + 2_y_ = 5

So, since *x* = 3:

3 × 3 + 2_y_ = 5

9 + 2_y_ = 5

Subtract 9 from both sides to get:

2_y_ = 5 – 9 = −4

Finally, divide by two to get:

*y** *= −4 ÷ 2 = −2

## Solving a System of Equations by Graphing

## Convert the Equations to Slope-Intercept Form

## Plot the Lines on a Graph

## Find the Point of Intersection

Solve systems of equations with minimal algebra by graphing each equation and looking for the *x* and *y* value where the lines intersect. Convert each equation to slope-intercept form (*y* = *mx* + *b*) first.

The first example equation is:

*x** *– *y* = 5

This can be converted easily. Add *y* to both sides and then subtract 5 from both sides to get:

*y** *= *x* – 5

Which has a slope of *m* = 1 and a *y*-intercept of *b* = −5.

The second equation is:

3_x_ + 2_y_ = 5

Subtract 3_x_ from both sides to get:

2_y_ = −3_x_ + 5

Then divide by 2 to get the slope-intercept form:

*y** *= −3_x_/2 + 5/2

So this has a slope of *m* = -3/2 and a *y*-intercept of *b* = 5/2.

Use the *y* intercept values and the slopes to plot both lines on a graph. The first equation crosses the *y* axis at *y* = −5, and the *y* value increases by 1 every time the *x* value increases by 1. This makes the line easy to draw.

The second equation crosses the *y* axis at 5/2 = 2.5. It slopes downwards, and the *y* value decreases by 1.5 every time the *x* value increases by 1. You can calculate the *y* value for any point on the *x* axis using the equation if it’s easier.

Locate the point where the lines intersect. This gives you both the *x* and *y* coordinates of the solution to the system of equations.