Solving Systems of Equations

Key Terms

0Solution0 A solution of a system of equations is an ordered pair

that satisfies each equation in the system.0Solving the system of equations

0 Finding the set of all solutions is called solving the system of equations.

The Method of Substitution

0Steps0 Solve one of the equations for one variable in terms of

the other.0 Substitute the expression found in step 1 into the other

equation to obtain an equation in one variable.0 Solve the equation in step 2.

Steps Continued

0Back-substitute the value(s) obtained in Step 3 into the expression obtained in Step 1 to find the value(s) of the other variable.

0Check that each solution satisfies both of the original equations.

ExampleOriginal

Step 1: Solve equation 1 for x in terms of y.

Equation 1

Equation 2

Step 2: Substitute y for x in equation 2 to obtain an equation in one variable.

Example ContinuedSolve the equation obtained in Step 2.

Back-substitute the value(s) obtained in Step 3 into the expression obtained in Step 1 to find the value(s) of the other variable.

Practice

−2 𝑥+𝑦=5

Practice

2 𝑥− 𝑦=−3How do you use substitution to solve systems of equations?

Word Problem

0Dorothy is 3 times as old as her sister. In 5 years she will be twice as old as her sister. How old are Dorothy and her sister now?

S= Dorothy’s Sister

In 5 years Dorothy will be D+5 and her sisters age will be S+5

𝐷=3𝑆Right now

Word Problem Continued

𝐷+5=2(𝑆+5)If she will be 2 times as old as her sister.

3𝑆+5=2 (𝑆+5 )Substitute from equation one D=3S

𝐷=3𝑆

The Method of Elimination

0The key step in the method of elimination is to obtain, for one of the variables, coefficients that differ only in sign so that adding the equations eliminates the variable.

The Method of Elimination

0To use the method of elimination to solve a system of two linear equations in x and y, perform the following steps.0 Obtain coefficients for x (or y) that differ only in sign by

multiplying all terms of one or both equations by suitable chosen constants.

0 Add the equations to eliminate one variable; solve the resulting equation.

The Method of Elimination Continued

0Back-substitute the value obtained in Step 2 into either of the original equations and solve for the other variable.

0Check your solution in both original equations.

Example

3 𝑥+5 𝑦=7Equation 1Equation 2

3 𝑥+5 𝑦=7Add the equations.

Example Continued

3 𝑥+5 (2 )=7Substitute 2 for y to solve for x

(−1,2)Solution

Check the Solution!!!

Example

5 𝑥+3 𝑦=9Equation 1Equation 2

Example Continued

4 (5 𝑥+3 𝑦=9 ) 20 𝑥+12 𝑦=36Multiply equation 1 by 4.Multiply Equation 2 by 3.Add equations.

Example Continued

20 𝑥+12 𝑦=36Add Equations

26 𝑥26

=7828

Solve for x.

Example Continued

2 𝑥−4 𝑦=14By back-substituting the value for x in equation 2, you can solve for y.

(3 ,−2)Solution

Graphical Interpretation of Solutions

0For a system of two linear equations in two variables, the number of solutions is one of the following.

Number of Solutions Graphical Interpretation

1. Exactly one solution. The two lines intersect at one point.

2. Infinitely many solutions. The two lines are coincident (identical).

3. No Solution The two lines are parallel.

Consistent or Inconsistent

0A system of linear equations is consistent if it has at least one solution. It is inconsistent if it has no solution.

Inconsistent System

Consistent System

Quick Quiz

1. The first step in solving a system of equations by the method of is to obtain coefficients for x (or y) that differ only in sign.

2. Two systems of equations that have the same solution set are called systems.

3. A system of linear equations that has at least one solution is called , whereas a system of linear equations that has no solution is called .