2x + 5y = 19
4x + 3y = 17
What are the values of x and y?

Step 1: Manipulate the Equations so that a term can cancel out
When we have a system of equations in which both equations are written in the form ax + by = c, we can solve the system using the method called ELIMINATION.
To use elimination we need to make sure that one of the pairs of variables cancels up to zero. Looking at the system we notice that the two x terms are 2x and 4x and the two y-terms are 5y and 3y. Neither of the pairs cancels up to zero so there is some work to do.
We need to decide what to multiply the equations by so that we can make one pair cancel up to zero. If we look at the x-terms we can multiply 2x by -2 and it will become 4x which if added to the other terms, 4x, will cancel up to zero. This means that we will multiply all terms in the top equation by -2:
-2 × (2x + 5y) = (19) × -2
4x + 3y = 24
So our system becomes,
-4x – 10y = -38
4x + 3y = 17
Step 2: Add the Equations and solve
Once we have one pair of the variable terms cancel up to zero we can add the equations to each other. We add x-term to x-term and y-term to y-term vertically.
-4x – 10y = -38
4x + 3y = 17
0x – 7y = -21
So our x-terms zero out (cancel out) giving us an equation that only has y so we can solve for y.
-7y = -21
Since -7 is being multiplied by x, we have to undo the multiplication by using division. Therefore we are going to divide both sides by -7.
(-7y/-7) = (-21/-7)
This simplifies to:
1y = 3
Since 1y and y are the same thing, our answer is:
y = 3
Step 3: Find the value of the other term
Now that we have the value of the y-term we can substitute it into either of the original equations to find the value of the x-term.
2x + 5y = 19
2x + 5(3) = 19
2x + 15 = 19
2x + 15 – 15 = 19 – 15
2x = 4
Since 2 is being multiplied by x, we have to undo the multiplication by using division. Therefore we are going to divide both sides by 2.
(2x/2) = (4/2)
This simplifies to:
1x = 2
Since 1x and x are the same thing, our answer is:
x = 2
So for this system our solution is x = 2 and y = 3.