Math CentralQuandaries & Queries


Question from Rose, a student:

I have been having trouble trying to figure out these three math problems , I need help breaking them down so I could understand them better please help.
1. y = 7 - x
2 x - y = 8

2. 8 x + 5 y = 1 8 4
x - y = 3

3. y + 2 x = 3
y + 2 x = 4

I can't figure out how to break them down in the right order.


There was a typo in your first problem, I hope my edit produced the correct problem.

For the first problem I would use the substitution method. You already know from the first problem that y = 7 - x. If you substitute 7 - x for y in the second equation then the second equation become an equation in x alone which you can solve. There is an example in my response to Amber a few days ago.

You can approach your second problem the same way. If you add y to each side of the second equation it becomes x = 3 + y. You can then substitute 3 + y for x in the first equation so that the first equation becomes an equation in y alone which you can solve. There is however a second method called the elimination method that you can use on any of your problems so I am going to illustrate it to you using your second problem,

8 x + 5 y = 1 8 4
x - y = 3

First multiply both sides of the second equation by 5. Then equations then become

8 x + 5 y = 1 8 4
5 x - 5 y = 15

Why did I multiply by 5? Look at the y terms. The coefficient of y in the first equation is 5 and the coefficient in the second equation is -5. Hence if I add the two equations the y terms will be eliminated. This is why I multiplied by 5.

Adding the two equations I get

8x + 5x = 184 + 15

Solve this equation for x and substitute the value you obtain for x into the equation x - y = 3 to find the value of y.

Finally look at your third problem. The coefficient of x in each of the equations is 1 so if you subtract the two equation you will eliminate the x terms. Subtracting the two equations I get

0 = -1.

What happened? Zero is not -1! This means that there is no solution to your third problem. Think of it graphically. Each of the two equations is a straight line and if you put them in the form y = mx + b you can see that they each have the same slope, -2, but different y-intercepts. Hence they are two different parallel lines and thus don't intersect.

I hope this helps,

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