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

**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:

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:

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:

Collect the like terms to get:

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

Dividing both sides by 5 gives:

So *y* = −2.

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

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

So *x* = 3 and *y* = −2.

#### Tips

### Solving a System of Equations by Elimination

Look at your equations to find a variable to remove:

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:

So

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:

So this means:

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

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:

So, since *x* = 3:

Subtract 9 from both sides to get:

Finally, divide by two to get:

### Solving a System of Equations by Graphing

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:

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

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

The second equation is:

Subtract 3*x* from both sides to get:

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

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.

References

About the Author

Lee Johnson is a freelance writer and science enthusiast, with a passion for distilling complex concepts into simple, digestible language. He's written about science for several websites including eHow UK and WiseGeek, mainly covering physics and astronomy. He was also a science blogger for Elements Behavioral Health's blog network for five years. He studied physics at the Open University and graduated in 2018.