Video: Solving Simultaneous Equations Graphically

By plotting the graphs of 𝑦 = −2𝑥 + 1 and 𝑦 = 𝑥 + 4, find the point that satisfies both equations simultaneously.

03:24

Video Transcript

By plotting the graphs of 𝑦 equals negative two 𝑥 plus one and 𝑦 equals 𝑥 plus four, find the point that satisfies both equations simultaneously.

We’re told in this problem that we need to approach it by plotting the graphs of these two equations. So, let’s consider two different methods for plotting straight-line graphs. In our first method, we’ll plot the graph of 𝑦 equals negative two 𝑥 plus one by comparing its equation to the general equation of a straight line in its slope–intercept form, 𝑦 equals 𝑚𝑥 plus 𝑏. We recall that, in this form, the value of 𝑚, that’s the coefficient of 𝑥, represents the slope of the line. So, the slope of the line we’re looking to plot is negative two. This means that for every one unit the line moves across to the right, it will move two units downwards.

We also recall that, in this general form, the value of 𝑏, the constant term, represents the 𝑦-intercept of the line. So in our equation, the 𝑦-intercept is positive one. This is the value at which the line intersects the 𝑦-axis. Let’s plot this line then. We know that it intercepts the 𝑦-axis at a value of one. As the slope is negative two, we know that if we move one unit across, we then need to move two units downwards. So, we can plot our next point at the coordinates one, negative one. We then move one across and two down again and plot our next point at two, negative three.

We can also go back the other way from our 𝑦-intercept. If we move one unit to the left, we need to move two units up. So, we can also plot a point with coordinates negative one, three. Joining all of these points together with a straight line, and we have our first line 𝑦 equals negative two 𝑥 plus one.

We’ll use a different method to plot the second line, which has equation 𝑦 equals 𝑥 plus four. This time, we’ll use a table of values. We’ll choose a range of different 𝑥-values. I’ve chosen the integer values from negative two to two. And we’ll then use the equation of the line to work out the corresponding 𝑦-values. For example, when 𝑥 is equal to zero, 𝑦 will be equal to zero plus four, which is four. When 𝑥 is equal to negative one, 𝑦 will be equal to negative one plus four or four minus one, which is three.

In the same way, we can then complete the rest of our table. We can then plot these points or at least those which will fit on the axes I’ve got here and then join them together with a straight line to give our second line 𝑦 equals 𝑥 plus four. Notice that the 𝑦-intercept of this line is four, and our line does indeed cross the 𝑦-axis at this point. And the slope is one, and our line does indeed have a slope of one. So, we’ve plotted the two lines. And now, we need to find the point that satisfies both equations simultaneously. This is the point that lies on both lines. It’s the coordinates of their point of intersection.

From our graph, we can see that the lines intersect at the point with coordinates negative one, three. As we we’re asked to find the point that satisfies both equations simultaneously, we’ll give our answer as a coordinate. So, our solution to the problem is the coordinate negative one, three.

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