3 Ways To Solve Systems Of Algebraic Equations Containing Two Variables

This method often uses fractions later on. You can try the elimination method below instead if you don’t like fractions.

4x = 8 - 2y (4x)/4 = (8/4) - (2y/4) x = 2 - ½y

You know that x = 2 - ½y. Your second equation, that you haven’t yet altered, is 5x + 3y = 9. In the second equation, replace x with “2 - ½y”: 5(2 - ½y) + 3y = 9.

5(2 - ½y) + 3y = 9 10 – (5/2)y + 3y = 9 10 – (5/2)y + (6/2)y = 9 (If you don’t understand this step, learn how to add fractions. This is often, but not always, necessary for this method. ) 10 + ½y = 9 ½y = -1 y = -2

You know that y = -2 One of the original equations is 4x + 2y = 8. (You can use either equation for this step. ) Plug in -2 instead of y: 4x + 2(-2) = 8. 4x - 4 = 8 4x = 12 x = 3

If you end up with an equation that has no variables and isn’t true (for instance, 3 = 5), the problem has no solution. (If you graphed both of the equations, you’d see they were parallel and never intersect. ) If you end up with an equation without variables that is true (such as 3 = 3), the problem has infinite solutions. The two equations are exactly equal to each other. (If you graphed the two equations, you’d see they were the same line. )

You have the system of equations 3x - y = 3 and -x + 2y = 4. Let’s change the first equation so that the y variable will cancel out. (You can choose x instead, and you’ll get the same answer in the end. ) The - y on the first equation needs to cancel with the + 2y in the second equation. We can make this happen by multiplying - y by 2. Multiply both sides of the first equation by 2, like this: 2(3x - y)=2(3), so 6x - 2y = 6. Now the - 2y will cancel out with the +2y in the second equation.

Your equations are 6x - 2y = 6 and -x + 2y = 4. Combine the left sides: 6x - 2y - x + 2y = ? Combine the right sides: 6x - 2y - x + 2y = 6 + 4.

You have 6x - 2y - x + 2y = 6 + 4. Group the x and y variables together: 6x - x - 2y + 2y = 6 + 4. Simplify: 5x = 10 Solve for x: (5x)/5 = 10/5, so x = 2.

You know that x = 2, and one of your original equations is 3x - y = 3. Plug in 2 instead of x: 3(2) - y = 3. Solve for y in the equation: 6 - y = 3 6 - y + y = 3 + y, so 6 = 3 + y 3 = y

If your combined equation has no variables and is not true (like 2 = 7), there is no solution that will work on both equations. (If you graph both equations, you’ll see they’re parallel and never cross. ) If your combined equation has no variables and is true (like 0 = 0), there are infinite solutions. The two equations are actually identical. (If you graph them, you’ll see that they’re the same line. )

The basic idea is to graph both equations, and find the point where they intersect. The x and y values at this point will give us the value of x and the value of y in the system of equations.

Your first equation is 2x + y = 5. Change this to y = -2x + 5. Your second equation is -3x + 6y = 0. Change this to 6y = 3x + 0, then simplify to y = ½x + 0. If both equations are identical, the entire line will be an “intersection”. Write infinite solutions.

If you don’t have graph paper, use a ruler to make sure the numbers are spaced precisely apart. If you are using large numbers or decimals, you may need to scale your graph differently. (For example, 10, 20, 30 or 0. 1, 0. 2, 0. 3 instead of 1, 2, 3).

In our examples from earlier, one line (y = -2x + 5) intercepts the y-axis at 5. The other (y = ½x + 0) intercepts at 0. (These are points (0,5) and (0,0) on the graph. ) Use different colored pens or pencils if possible for the two lines.

In our example, the line y = -2x + 5 has a slope of -2. At x = 1, the line moves down 2 from the point at x = 0. Draw the line segment between (0,5) and (1,3). The line y = ½x + 0 has a slope of ½. At x = 1, the line moves up ½ from the point at x=0. Draw the line segment between (0,0) and (1,½). If the lines have the same slope, the lines will never intersect, so there is no answer to the system of equations. Write no solution.

If the lines are moving toward each other, keep plotting points in that direction. If the lines are moving away from each other, move back and plot points in the other direction, starting at x = -1. If the lines are nowhere near each other, try jumping ahead and plotting more distant points, such as at x = 10.