Find the number of solutions to the following system of equations: 𝑦 plus two 𝑥 equals four and two 𝑦 plus four 𝑥 equals eight. The options are no solutions, one solution, or infinite solutions.
A solution to this pair of linear equations, if it exists, will be a pair of 𝑥𝑦-values that satisfy both equations simultaneously. We notice that in the first equation, the coefficient of 𝑦 is one, which means it’s relatively straightforward to rearrange this equation to make 𝑦 the subject. We can therefore use the substitution method to try to solve this pair of equations and hence determine the number of solutions. We make 𝑦 the subject of the first equation by subtracting two 𝑥 from each side. And we find that 𝑦 is equal to negative two 𝑥 plus four.
We can then substitute this expression for 𝑦 into our second equation. That gives two multiplied by negative two 𝑥 plus four plus four 𝑥 is equal to eight. To simplify this equation, we’ll first distribute the parentheses on the left-hand side. We have negative four 𝑥 plus eight plus four 𝑥 is equal to eight. Now on the left-hand side, the negative four 𝑥 and the positive four 𝑥 will cancel each other out. And we’re left with simply eight is equal to eight. Now this may seem a little strange, but let’s consider what this is telling us.
After we perform this substitution, the equation has reduced to an equation which is independent of 𝑦 and is also independent of 𝑥. Eight is always equal to eight. So this equation is always true. This tells us that there are, in fact, infinitely many solutions to this system of linear equations. Another way of describing this is that any ordered pair of 𝑥𝑦-values that satisfies the first equation will also satisfy the second. We can see why this is the case if we examine the second equation in more detail. All of the coefficients in this equation are even numbers, so we can divide the entire equation by two. When we do this, we obtain 𝑦 plus two 𝑥 is equal to four.
But look, this is exactly the same as the first equation. So it makes sense that there are an infinite number of solutions, as any ordered pair of 𝑥𝑦-values that satisfy the first equation will automatically satisfy the second. If we were to attempt to solve these equations graphically, then we’d need to plot the line 𝑦 plus two 𝑥 equals four or 𝑦 equals negative two 𝑥 plus four twice. The two lines representing these two equations would be what we call coincident. One line lies directly on top of the other. And so any point on the line will be a solution to the pair of simultaneous equations. But as this line extends infinitely in both directions, there are an infinite number of points on the line and so an infinite number of solutions to the system of equations.
Our answer then is (c), there are infinitely many solutions to this system of equations.