Solving Systems of Linear Equations

Definition: A system of equations is a set of two or more equations with a same unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system.

Example: Solve 3y – 2x = 11   y + 2x = 9

Solution:

By Elimination (also known as addition)

Step 1: Choose to eliminate the variable whose numerical coefficient (without considering the sign) is the same. (If not the case for either variable, then muliply each equation by a factor that will cause the coefficients to be the same. For example, if we wanted to eliminate y then we would multiply the first equation by 1 and the second by 3.)

Step 2: If the signs of the coefficients of the variables are not the same then add the equations, if they are, subtract them.

3y – 2x = 11
+(y + 2x = 9)
4y = 20

Step 3: Solve the resulting equation.
4y/4 = 20/4
y = 5

Step 4: Use the obtained value (and either equation) to solve for the other unknown.
5 + 2x = 9
5 – 5 + 2x = 9 – 5
2x = 4
2x/2 = 4/2
x = 2

Solution (2, 5)

By Substitution

Step 1: Make one of the unknowns the subject of either equation.
Using y + 2x = 9, we get y = 9 – 2x.

Step 2: Substitute 9 – 2x for y in the other equation.
3(9-2x) – 2x = 11

Step 3: Simplify and solve.
27 – 6x – 2x = 11
27 – 8x = 11
27 – 27 – 8x = 11 – 27
-8x = -16
-8x/-8 = -16/-8
x = 2

Step 4: Use the obtained value (and either equation) to solve for the other unknown.
y = 9 – 2(2)
y = 9 – 4
y = 5

Solution: (2, 5)

By Graphing

Step 1: Create a table of values with at least two points for both equations.

Step 2: Plot the points and complete the graphs.

Step 3: The point of intersection is the solution. Check out how a Yup tutor helps a student solve a system of linear equations!

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