# Lesson Video: Solving Systems of Linear Equations Graphically Mathematics • 8th Grade

In this video, we will learn how to solve a system of two linear equations by considering their graphs and identifying the point of intersection.

15:13

### Video Transcript

In this video, we will learn how to solve a system of two linear equations by considering their graphs and identifying their point of intersection.

We recall first of all that a linear equation is one in which the highest power of each variable that appears is one and there are no terms in which the variables are multiplied together. For example, the equation two 𝑥 plus 𝑦 equals six is a linear equation.

A system of two linear equations is simply a pair of two such equations. For example, if we also had the 𝑥 plus 𝑦 equals two equation, we now have a system of linear equations, sometimes known as a pair of simultaneous equations. There are many different methods that can be used to solve such systems of equations, but in this video, we’re focusing on the graphical method. As a result, the two letters we use to represent our variables will often be 𝑥 and 𝑦, but this doesn’t have to be the case.

The solution to a system of two linear equations can be found by plotting a graph of the two straight lines represented by these equations and then identifying the coordinates of their point of intersection. This is because this point lies on both lines and therefore satisfies both equations simultaneously.

In our first example, we’ll review how to find an equation of a straight line from its graph. This will in turn enable us to identify the system of linear equations that can be solved using a given graph.

Which of the following sets of simultaneous equations could be solved using the given graph? (A) 𝑦 equals two 𝑥 minus four, 𝑦 equals 𝑥 plus five. (B) 𝑦 equals negative four 𝑥 plus two, 𝑦 equals five 𝑥 minus one. (C) 𝑦 equals two 𝑥 minus four, 𝑦 equals negative 𝑥 plus five. (D) 𝑦 equals two 𝑥 plus four, 𝑦 equals negative 𝑥 plus five. Or (E) 𝑦 equals negative four 𝑥 plus two, 𝑦 equals five 𝑥 plus one.

We’ve been given a graph of two straight lines. Let’s name them 𝑙 one and 𝑙 two. We are asked to determine which pair of simultaneous equations we could solve using this graph. This means that we need to determine the equations of the two straight lines.

In order to do this, we’ll recall the general form of a straight line in its slope–intercept form, 𝑦 equals 𝑚𝑥 plus 𝑏. And we recall that the coefficient of 𝑥, that’s 𝑚, gives the slope of the line. And the constant term, that’s 𝑏, gives the 𝑦-intercept of the graph. That’s the 𝑦-value at which the line intercepts the 𝑦-axis.

We can determine both of these values from the diagram. First, line 𝑙 one has a 𝑦-intercept of five, and line 𝑙 two has a 𝑦-intercept of negative four. Next, we determine the slope of each line by using the fact that a line passing through two coordinate points — 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two — has the slope 𝑚 equals 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. 𝑙 one passes through the points zero, five and one, four. We could have selected any two points from this line. But it makes our work easier if we choose a second point near the 𝑦-intercept.

The first line has a slope of 𝑚 equals four minus five over one minus zero. So, the slope of 𝑙 one is negative one. The second line 𝑙 two passes through the points zero, negative four and one, negative two. Its slope is calculated by 𝑚 equals negative four minus negative two over zero minus one. So, the slope of 𝑙 two is positive two. Hence, the line 𝑙 one has 𝑏 equal to five and 𝑚 equal to negative one. Its slope–intercept equation is 𝑦 equals negative 𝑥 plus five, which is one of the equations given in option (C) and in option (D).

Line 𝑙 two has 𝑏 equal to negative four and 𝑚 equal to two. So, its equation is 𝑦 equals two 𝑥 minus four. This gives us the system of equations 𝑦 equals two 𝑥 minus four, 𝑦 equals negative 𝑥 plus five. The solutions to this system of linear equations given by the coordinates of the point of intersection are 𝑥 equals three and 𝑦 equals two. The given graph shows the system of simultaneous linear equations in option (C): 𝑦 equals two 𝑥 minus four, 𝑦 equals negative 𝑥 plus five.

Let’s now consider a second example.

Use the shown graph to solve the given simultaneous equations: 𝑦 equals four 𝑥 minus two, 𝑦 equals negative 𝑥 plus three.

We recall that the solution to a system of equations is given by the coordinates of the point of intersection between the graphs of all of the equations. This means the coordinates of the point of intersection between the two lines tells us the solution to the simultaneous equations. We see that the 𝑥-coordinate of this point is one and the 𝑦-coordinate is two. This tells us 𝑥 equals one and 𝑦 equals two is a solution to the simultaneous equations.

We can verify this by substituting these values into the equations. If we substitute 𝑥 equals one into the first equation, we get 𝑦 equals four times one minus two equals two, which agrees with our solution. Similarly, if we substitute 𝑥 equals one into the second equation, we get 𝑦 equals negative one plus three equals two, which also agrees with our solution. Since both equations hold true, this verifies the solution. Since this is the only point of intersection, it is the only solution to the simultaneous equations. Therefore, the only solution is 𝑥 equals one and 𝑦 equals two.

In our next example, we’ll need to plot the graphs of the two equations that we wish to solve ourselves. So, we’ll remind ourselves of some of the key methods for doing this.

By plotting the graphs of 𝑦 equals 𝑥 minus one and 𝑦 equals five 𝑥 plus seven, find the point that satisfies both equations simultaneously.

We recall that if the pair of coordinates 𝑥 and 𝑦 satisfy both equations simultaneously, then the point must lie on the graphs of both equations. Hence, it is their point of intersection, and so it is a solution to the system of equations. Therefore, we can find the solutions to this system by finding the coordinates of the points of intersection. We do this by sketching both graphs on the same coordinate plane. We note that both lines are of the form 𝑦 equals 𝑚𝑥 plus 𝑏. We recall that this line will have a 𝑦-intercept of 𝑏 and a slope of 𝑚, provided 𝑚 does not equal zero.

We can interpret the important information about each line by referring to this formula. First, the line 𝑦 equals 𝑥 minus one will have a 𝑦-intercept of negative one. We know that the coefficient of 𝑥 is one, so the slope of the first line must be one. We recall that the slope of a line is the change in 𝑦 with respect to the change in 𝑥. Therefore, a slope of one represents an increase of one in 𝑦 and an increase of one in 𝑥. Following the change of coordinates, up one and right one, we get another coordinate point on the line: one, zero. Connecting these points allows us to sketch the line with a slope of one and a 𝑦-intercept of negative one.

Moving on to the second linear equation, we have a 𝑦-intercept of seven and a slope of five. This means the change in the 𝑦-coordinate is five and the change in the 𝑥-coordinate is one. So, we start at the 𝑦-intercept then move up five and right one. However, because of the position of the 𝑦-intercept near the edge of our coordinate plane, we can reverse these directions as needed to find a point on the line to the left of zero, seven. So, we will move down five and left one instead. We found a point on the line with coordinates 𝑥 equals negative one and 𝑦 equals two.

Finally, we connect these points to sketch the line 𝑦 equals five 𝑥 plus seven. We can see that both lines contain the point negative two, negative three. This is the point of intersection, meaning 𝑥 equals negative two and 𝑦 equals negative three satisfy both equations.

We can verify that these coordinates satisfy both equations by substituting 𝑥 equals negative two. We note that both equations give 𝑦 equals negative three. Since both equations hold true, this verifies the solution we found by graphing. Hence, 𝑥 equals negative two and 𝑦 equals negative three satisfies both equations. And we can say that the point that satisfies both equations is negative two, negative three.

Plot the graphs of the simultaneous equations 𝑦 equals two 𝑥 plus seven, 𝑦 equals two 𝑥 minus four, and then solve the system.

We recall that the points of intersection of the graphs of both equations will tell us the solutions to the system of equations. This means we can solve the system by sketching both equations on one coordinate plane. Since these are linear equations given in slope–intercept form, we will plot their graphs using their slope and 𝑦-intercepts.

We’ll begin with the first equation, 𝑦 equals two 𝑥 plus seven. We see that two is the coefficient of 𝑥 and seven is the constant. That means that two is the slope and seven is the 𝑦-intercept. So, we plot the 𝑦-intercept of the first line at seven. The second line has the same slope as the first line but a different 𝑦-intercept, negative four.

Now, we recall that slope is the change in 𝑦 with respect to the change in 𝑥. Therefore, a slope of two can be interpreted as adding two to the 𝑦-coordinate and adding one to the 𝑥-coordinate. To find another point on the second line, we simply move up two and right one. Then, to plot the graph of the line 𝑦 equals two 𝑥 minus four, we connect the 𝑦-intercept to the new point with a straight line. This is the line given by the equation 𝑦 equals two 𝑥 minus four.

Since the first line has the same slope as the second line, we can plot its graph by sketching a parallel line through the 𝑦-intercept of seven. This is the line given by the equation 𝑦 equals two 𝑥 plus seven. Seeing that these lines run parallel, we know that there are no points of intersection, so this system has no solutions.

In fact, we could have saved ourselves the effort of drawing the graphs by just carefully examining the equations of the two lines. We can see that both lines have the same slope, but distinct 𝑦-intercepts. This tells us that these lines are parallel; they have the same slope. And they are distinct; they pass through different 𝑦-intercepts. Hence, the lines do not intersect and the system has no solutions.

In our examples so far, we’ve seen two possibilities. Firstly, the lines could intersect at a single point, in which case there is one solution to the system of linear equations. We call this an independent system of linear equations. Secondly, the lines could be parallel, if they have the same slope and distinct 𝑦-intercepts. In this case, there is no solution to the system of simultaneous equations, as the two lines will never intersect. We call this an inconsistent system of equations. Systems with at least one solution are considered consistent, such as in the first case.

There is in fact a third option. Suppose we were asked to solve the system of equations 𝑦 equals two 𝑥 minus four and four 𝑥 minus two 𝑦 equals eight. If we were to plot these graphs, we can see that these two equations describe the exact same line. This is because if we write the second equation in slope–intercept form, we get 𝑦 equals two 𝑥 minus four. This means the second equation is an equivalent way of writing the equation of the first line. In this case, the lines are described as being coincident. One line lies exactly on top of the other. And every single point on this infinitely long line will therefore satisfy the system of linear equations. We therefore say that there are infinitely many solutions. We call this a dependent system of linear equations, which is also considered consistent because this type of system has solutions.

So, these are the three options when solving a pair of linear, simultaneous equations graphically.

Let’s review the key points we’ve seen in this video. Firstly, we saw that systems of linear equations can be solved by plotting their graphs and identifying the coordinate points of their intersection. However, we also saw that not all straight lines intersect. The three possibilities are that the lines intersect at one point, meaning there is one solution consisting of an 𝑥-coordinate and 𝑦-coordinate. The two lines are parallel. They never intersect, so there are no solutions. Or the two lines are coincident, in which case there are infinitely many solutions to the system of linear equations.