Video: Solving Simultaneous Equations Graphically

Plot the graphs of the simultaneous equations 𝑦 = 2𝑥 + 7, 𝑦 = 2𝑥 − 4, and then solve the system.

02:07

Video Transcript

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

We’ll plot each of these graphs by comparing their equations with the general equation of a straight line in its slope–intercept form, 𝑦 equals 𝑚𝑥 plus 𝑏. The first line 𝑦 equals two 𝑥 plus seven has a slope of two and a 𝑦-intercept of seven. We can plot this line by first plotting the 𝑦-intercept. And then for every one unit we go to the right, we go two units up. In this case though, we’ll go the other way. For every one unit we go to the left, we go two units down. So that gives a point at negative one, five; and then a point at negative two, three; and a point at negative three, one. We can then join all of these points up to give our first plotted line.

In the same way, we see that the line 𝑦 equals two 𝑥 minus four has a slope of two and a 𝑦-intercept of negative four. We can plot the 𝑦-intercept. And then, we go one unit to the right and two units up, one unit right and two units up again, and again. And then, we join these points up to give our second line. So, we’ve plotted the graphs of these two simultaneous equations. But now, we’re asked to solve the system, which means we’re looking for the point of intersection of these two lines.

Now, our two lines don’t intersect on the graph I’ve drawn. Does this mean that I’ve just chosen the wrong ranges for the 𝑥- and 𝑦-axes and they would intersect if I’d chosen a larger range? Well, the answer to that is no, these two lines will actually never intersect. And the reason for this is because they are parallel lines. They both have the same slope of two. We know that parallel lines will never meet. And therefore, these two lines will have no point of intersection. So, in fact, there are no solutions to this system of simultaneous equations. And our reasoning for this is that the two equations represent parallel lines.

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