Use the shown graph to solve the given simultaneous equations: 𝑦 equals negative two 𝑥 plus two and 𝑦 equals three 𝑥 plus two.
To solve, we need to look at the lines and decide where they cross, what is that point. As we can see, they cross at the point zero, two. So this means 𝑥 equals zero and 𝑦 equals two. This would be our final answer, but let’s go ahead and just double-check, add our equations, and see which equation goes with which line.
When you have an equation, you have 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is your slope and 𝑏 is your 𝑦 intercept. So the slope kind of tells us how steep the line is, and the 𝑦-intercept tells us where it crosses the 𝑦-axis.
So looking at our first equation, 𝑦 equals negative two 𝑥 plus two, we have a slope of negative two, so we should be going down left to right, and then a 𝑦-intercept at two, so it crosses it at two and then it should be going down.
Essentially, slope is rise over run, so here we can see we’re rising negative two, so it means we’re going down two, and then we go positive one, so to the right one. So from two, we go down two and then we go right one, and we’re here. And now if we do it again, down two and right one, so we can see this first equation goes with the green line.
The next equation has a slope of positive three, so it should be increasing or going up left to right and crossing the 𝑦-axis at two. So starting at two, we need to go up three and then right one, so here we’ve gone up three and then right one. So that equation goes with the yellow line.
So again, we’ve located which equation goes with which line, and to solve it’s where they cross, and they cross at the point zero, two. So our final answer will be 𝑥 equals zero and 𝑦 equals two.