Find the number of solutions to the following system of equations: 𝑦 equals 𝑥 plus one and 𝑦 equals negative 𝑥 plus nine. The options are zero solutions, one solution, or infinite solutions.
We’re asked to find the number of solutions to the given system of equations. Now, a solution to this system of equations, if it exists, will comprise a pair of 𝑥- and 𝑦-values that satisfy both equations simultaneously. We’ll consider two approaches to this question, the first being to attempt to solve this system of equations.
We note that 𝑦 is the subject of both equations. In each case, we have an expression for 𝑦 in terms of the other variable, which is 𝑥. For a pair of 𝑥𝑦-values to satisfy this system of equations, then the 𝑦-value in the two equations must be equal when we substitute the corresponding 𝑥-value.
To see if there are many solutions to this system of equations then, we can equate the two expressions we’ve been given for 𝑦 in terms of 𝑥, giving the equation 𝑥 plus one is equal to negative 𝑥 plus nine. To attempt to solve this equation, we want to collect all of the 𝑥-terms on the same side. So we begin by adding 𝑥 to each side of the equation, which gives two 𝑥 plus one is equal to nine. We can then subtract one from each side of the equation, giving two 𝑥 is equal to eight, and then divide both sides of the equation by two to give 𝑥 is equal to four.
As we were able to solve this equation, this means that the 𝑦-value in the two equations will be the same when 𝑥 is equal to four. But let’s just confirm this. When 𝑥 is equal to four, the value of 𝑦 in the first equation will be four plus one, which is equal to five. And in the second equation, when 𝑥 is equal to four, 𝑦 is equal to negative four plus nine. That’s nine minus four, which is also equal to five. This confirms then that the pair of values 𝑥 equals four and 𝑦 equals five are a solution to the given system of equations. They are the only solution we found, so our answer is (b).
Let’s now consider another method that we could use to answer this question using our knowledge of straight-line graphs. We know that when the equation of a straight line is given in the slope–intercept form, 𝑦 equals 𝑚𝑥 plus 𝑏, then the value of 𝑚, the coefficient of 𝑥, represents the slope of that line and the value of 𝑏, the constant term, represents its 𝑦-intercept. We can therefore determine the slope and 𝑦-intercept of each of these lines and sketch them. The coefficient of 𝑥 in the first equation is one. So this is its slope. And the coefficient of 𝑥 in the second equation is negative one. In the first equation, the 𝑦-intercept is positive one. And in the second equation, the 𝑦-intercept is positive nine.
We can then sketch the lines represented by these two equations on the same set of axes, firstly the line 𝑦 equals 𝑥 plus one, with a slope and 𝑦-intercept of positive one, then the line represented by the equation 𝑦 equals negative 𝑥 plus nine, with a slope of negative one — so it slopes downwards from left to right — and a 𝑦-intercept of nine. We can see that these two lines intersect at a single point, indicating one solution to this system of equations.
The value of 𝑥 will be the 𝑥-coordinate of this point, and the value of 𝑦 will be the 𝑦-coordinate. We can see that the 𝑥-coordinate of this point is positive. So this is consistent with our 𝑥-value of four. And the 𝑦-coordinate of this point is somewhere between one and nine, which is consistent with our 𝑦-value of five.
So either by explicitly solving this system of equations or using our knowledge of straight lines to sketch their graphs, we found that there is one solution — that’s one pair of 𝑥𝑦-values — that satisfies the given system of equations.