Question Video: Finding the Number of Solutions to a System of Simultaneous Equations Mathematics • 8th Grade

Find the number of solutions to the following system of equations: π¦ = π₯ + 1, π¦ = βπ₯ + 9. [A] 0 solutions [B] 1 solution [C] Infinite solutions

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Video Transcript

Find the number of solutions to the following system of equations: π¦ equals π₯ plus one and π¦ equals negative π₯ plus nine. The options are zero solutions, one solution, or infinite solutions.

Weβ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 π₯- and π¦-values that satisfy both equations simultaneously. Weβll consider two approaches to this question, the first being to attempt to solve this system of equations.

We note that π¦ is the subject of both equations. In each case, we have an expression for π¦ in terms of the other variable, which is π₯. For a pair of π₯π¦-values to satisfy this system of equations, then the π¦-value in the two equations must be equal when we substitute the corresponding π₯-value.

To see if there are many solutions to this system of equations then, we can equate the two expressions weβve been given for π¦ in terms of π₯, giving the equation π₯ plus one is equal to negative π₯ plus nine. To attempt to solve this equation, we want to collect all of the π₯-terms on the same side. So we begin by adding π₯ to each side of the equation, which gives two π₯ plus one is equal to nine. We can then subtract one from each side of the equation, giving two π₯ is equal to eight, and then divide both sides of the equation by two to give π₯ is equal to four.

As we were able to solve this equation, this means that the π¦-value in the two equations will be the same when π₯ is equal to four. But letβs just confirm this. When π₯ is equal to four, the value of π¦ in the first equation will be four plus one, which is equal to five. And in the second equation, when π₯ is equal to four, π¦ is equal to negative four plus nine. Thatβs nine minus four, which is also equal to five. This confirms then that the pair of values π₯ equals four and π¦ equals five are a solution to the given system of equations. They are the only solution we found, so our answer is (b).

Letβ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 slopeβintercept form, π¦ equals ππ₯ plus π, then the value of π, the coefficient of π₯, represents the slope of that line and the value of π, the constant term, represents its π¦-intercept. We can therefore determine the slope and π¦-intercept of each of these lines and sketch them. The coefficient of π₯ in the first equation is one. So this is its slope. And the coefficient of π₯ in the second equation is negative one. In the first equation, the π¦-intercept is positive one. And in the second equation, the π¦-intercept is positive nine.

We can then sketch the lines represented by these two equations on the same set of axes, firstly the line π¦ equals π₯ plus one, with a slope and π¦-intercept of positive one, then the line represented by the equation π¦ equals negative π₯ plus nine, with a slope of negative one β so it slopes downwards from left to right β and a π¦-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 π₯ will be the π₯-coordinate of this point, and the value of π¦ will be the π¦-coordinate. We can see that the π₯-coordinate of this point is positive. So this is consistent with our π₯-value of four. And the π¦-coordinate of this point is somewhere between one and nine, which is consistent with our π¦-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 β thatβs one pair of π₯π¦-values β that satisfies the given system of equations.