# How to Solve a System of Linear Equations

# A System of Linear Equations

Let’s use the following system of Linear Equations for this post:

### y = 2x + 1

### y = – x + 7

### Task (solve the system of linear equations):

- Find the x- and y-intercepts
- Graph the equations
- Find the slopes
- Find number of solutions

The way to graph linear functions (basically just lines) is to find two things:

- The x-intercept (plug in 0 for y and solve for x)
- The y-intercept (plug in 0 for x and solve for y)

Let’s start with y = 2x + 1.

- To find the x-intercept, let’s plug in 0 for y and solve for x.

0 = 2x + 1

So, 2x = – 1 (sent the 1 to the other side)

Therefore, x = -1/2

- To find the y-intercept, let’s plug in 0 for x and solve for y.

y = 2(0) + 1

So, y = 1

Now, let’s put these two points on the graph and draw a line between them:

Let’s do the same thing for the second equation: y = – x + 7.

- To find the x-intercept, let’s plug in 0 for y and solve for x.

0 = -x + 7

So, x = 7 (sent the x to the other side)

- To find the y-intercept, let’s plug in 0 for x and solve for y.

Y = -(0)+ 7

So, y = 7

Now let’s graph this one on the same graph (plot the x- and y-intercepts and draw a line between):

**Hint: **You can graph linear equations using google. Just type in the equation in the google search bar. For example, just type in “Graph y=2x+1” and press enter. You can enter multiple functions by placing a comma between the functions. For example, “Graph y=2x+1, y=-x+7” and hit enter. Use this method to check if your answers are correct.

__To find the Slopes:__

__To find the Slopes:__

The standard form for an equation of a line is, where m is the slope and b is the y-intercept. So, if a question already gives you an equation in this form, all you have to do to get the slope is read the value in front of the x. For example, in y = 2x + 1 the slope is clearly 2.

*So the slopes of the lines for y = 2x + 1 and y = – x + 7 would be 2 and -1, respectively.*

If the equation is not in the standard form, all you have to do is move around the variables to put it in standard form. For example:

Now, we can see that 2/3 is the slope.

__Number of Solutions__

__Number of Solutions__

For a set of linear equations, a number of solutions just means: “* Do these lines ever meet each other at any point?*”

There are three possible solutions:

- 1 Solution: They meet at exactly one point.

- Infinite Solutions: The lines are exactly the same (even though the equations look different, when you plugin points, they have the same values). Usually these equations are just multiples of each other.

So, for our system of linear equations, clearly, the number of solutions is just 1 since they meet or intersect at one point. You can graph all the equations using google like I mentioned earlier to see what kind of graphs you get and you can tell the number of solutions easily from that. Also, you know that if two equations have the same slope, they are either parallel or the same line – so, either no solution or infinite solutions.

There are, of course, many ways to solve **a system of linear equations**. You may even be given a word problem and have to set up the initial equations yourself. Or, you may be asked to **solve the system of linear equations by method of substitution or elimination**. Try to become familiar with all of these methods and no matter what scenario or type of questions you’re given, you’ll be able to solve it easily.

PS: Check out and download **this great study note** with practice questions for systems of linear equations!