Download 4.1 Solving Systems of Linear Equations by Graphing

4.1 Solving Systems of Linear Equations by Graphing Learning Objectives: 1. Determine if an ordered pair is a solution of a system of linear equations. 2. Solve a system of linear equations by graphing. 3. Solve applied problems involving systems of linear equations. 1. Determine if an ordered pair is a solution of a system of linear equations Definition: 1. A system of linear equations—is a grouping of two or more linear equations where each equation contains one or more variables. ⎧x − y + z = 5
⎧4 x − 3 y = 12
⎪
2. ⎨ y + 2 x − z = 2 Example. 1. ⎨
⎩− 2 x − y = 0
⎪x + 2z = 0
⎩
2. A solution of a system of equations—is an ordered pair (x y ) that gives true to both equations. (is the point where the graphs intersect) Example 1. Is (3, 9 ) a solution of the given system? ⎧5 x − 2 y = −3
⎨
⎩ y = 3x
2. Solve a system of linear equations by graphing Three types of the System of Equations. 1. Consistent with Independent Equation (independent system) • Two lines intersect at one point ( x, y ) . • Has one solution ( x, y ) . • m1 ≠ m2 • When solve the system, get x = a number, y = a number. 2. Inconsistent System • Two lines are parallel. • Has no solution. • m1 = m2 and b1 ≠ b2 • When solve, get false statement. 3. Consistent with Dependent Equation (Dependent System) • Two lines lie on top of the others (same line). • Has infinitely many solutions. • m1 = m2 and b1 = b2 • When solve the system, get true statement. 1 y
x
y
x
y
x
Steps to solve system of linear equations by graphing: 1. Write an equation as the slope‐intercept form and graph each equation separately. 2. Identify type of systems (consistent, inconsistent, or dependent). 3. State number of solution (one solution, infinitely many solutions or no solution). Example 2. Solve, graph, label type of system and state number of solution. Label at least two y points for each graph. 10
⎧x + 2 y = 4
1. ⎨
⎩2 x − 4 y = 0
⎧ y − 3x = 3
2. ⎨
⎩2 y − 6 x = −8
2 -10
10
x
–10
y 10
-10
10
–10
x
1
y ⎧2
⎪⎪ 3 x + 3 y = 1
3. ⎨
10
⎪1 x + 1 y = 3
⎪⎩ 2
4
4
x
-10
10
–10
3. Solve applied problems involving systems of linear equations Example 3. Phone Charges A long‐distance phone service provider has two different long‐distance phone plans. Plan A charges a monthly fee of $ 40 plus $0.05 per minute. Plan B charges a monthly fee of 10$ plus $0.20 per minute. a. Write a linear equation for the cost of Plan A for x miles and Write a linear equation for the cost of Plan A for x miles b. Determine the number minutes, x, for which the cost, y, of each plan will be the same. 3 b. If you typically use 100 long‐distance minutes each month, which plan should you choose? c. Graph both equations on the grid provided and label at least three points for each equation. y 4