1. 7.3 Systems of Linear Equations in Two Variables

2. Objectives
Decide whether an ordered pair is a solution of a linear system.
Solve linear systems by graphing.
Solve linear systems by substitution.
Solve linear systems by addition.
Identify systems that do not have exactly one ordered-pair solution.
Solve problems using systems of linear equations.

3. System of Linear Equations in 2 variables
System of Linear Equations: linear equations
E.g., 2x – 3y = x + y = 4
Solution to a System of Equations: -- an ordered pair that satisfy both equations
E.g., (1, 2), or, x = 1 and y = 2 is a solution to the system
Check 2x – 3y = x + y = 4 2(1) – 3(2) = (1) + 2 = 4 2 – 6 = -4 true = 4 true

4. Graphical Interpretation of the Solution
2x + y = 4 y = -2x + 4
2x – 3y = -4 2x + 4 = 3y y = (2/3)x + (4/3)
(1, 2)

5. Example
Solve by graphing the system of equations. x + 2y = 2 and x – 2y = 6
Find x-intercept and y-intercept for both equations x + 2y = x – 2y = when x = 0, when x = 0, y = – 2y = 6 y = y = when y = when y = 0, x + 2(0) = x – 2(0) = 6 x = x = (0, 1) -- y-intercept (0, -3) -- y-intercept (2, 0) -- x-intercept (6, 0) -- x-intercept

6. Example (cont.) Lines intersect at (4, -1).
Check: x + 2y = (-1) = 2 4 – 2 = true x – 2y = 6 4 – 2(-1) = = true

7. Solving Equation System by substitution
Solve: y = -x – 1 4x – 3y = 24
Solution Substitute the expression for y in the first equation for y in the second equation. 4x – 3y = y = -x x – 3(-x – 1) = y = -(3) x + 3x + 3 = y = -4 7x = x = Solution: (3, -4)

8. Solving A linear System by Substitution
Solve y = -x – 1 4x – 3y = 24
This gives us: 4x – 3(−x – 1) = 24. Solving for x, we get: x = 3
Substitute x value back in the first equation. y = -(3) – 1
This gives us: y = -4
Solution: (3, -4)

9. Example Solve the linear system. -4x + y = -11 2x – 3y = 3 Solution
Using one equation, express y in terms of x. -4x + y = -11 y = 4x - 11
Substitute this in the second equation. 2x – 3y = 3 2x – 3(4x – 11) = 3
Solve for x 2x – 12x + 33 = x = -30 x = 3

10. Example (cont.)
Substitute this value of x in the first equation and solve for y. y = 4x – 11 y = 4(3) – 11 y = 12 – 11 y = 1
Solution: (3, 1)

11. Your turn Solve the linear system Solution What does this mean?
y = 2x + 7 2x – y = -5
Solution
2x – y = -5 2x – (2x + 7) = -5 2x – 2x + 7 = -5 7 = -5
What does this mean?
No combination of (x, y) can make this true—no solution
Check the slopes of the 2 lines.
Line 1: y = 2x + 7 Line 2: y = 2x + 5

12. Solving a linear system by addition
Solve the system by addition (elimination). 3y = 14 + x x + 22 = 5y
Arrange in general form. -x + 3y = 14 x – 5y = -22
Eliminate one column by adding column by column. -x + 3y = 14 x – 5y = y = -8
Solve one variable at a time y = 4

13. Substitute for y in either equation
Substitute for y in either equation. 3y = 14 + x 3(4) = 14 + x = 14 + x x = -2
Solution: (-2, 4)

14. Solving a linear system by addition
Solve: 3x + 2y = 48 9x – 8y = -24
The idea is to eliminate either the x column or the y column and add the two equations.
4(3x + 2y) = 4(48) 9x – 8y = -24
12x + 8y = x – 8y = x = 168
x = 8

15. Solving a linear system by addition
Substitute this value of x in either equation and solve for y. 3x + 2y = 48 3(8) + 2y = y = 48 2y = 24 y = 12
Solution: (8, 12)

16. Your Turn
Solve the system of equation using the addition method. 3x + y = 7 x + 2y = 4
3x + y = 7 -3(x + 2y) = -3(4) 3x + y = 7 -3x – 6y = y = - 5
y = x + 2y = 4 → x + 2(1) = 4 →x = 2 Solution: (2, 1)

17. A system with no solution
Solve: 4x + 6y = 12 6x + 9y = 12
Using the addition method, 4x + 6y = 12 6x + 9y = = false. There is no solution to the system.
Multiply by 3
Multiply by -2

18. A system with infinitely many solutions
Solve: y = 3x – 2 15x – 5y = 10
Using the substitution method, 15x – 5y = x – 5(3x – 2) = x – 15x + 10 = = This is true for any (x, y) pairs.
Thus, there is an infinitely number of solutions.

19. Special Cases Number of Solutions Graphically
One ordered-pair solution
Two lines intersect at one point.
No solution
Two lines are parallel.
Infinitely many solutions
Two lines are the same line.

20. Modeling with systems of equations
Suppose a company produces and sells x iGizmos.
Revenue function: R(x) = (price per unit sold)x
Cost function: C(x) = fixed cost (cost per unit produced)x
Break-even point: intersection of R(x) and C(x)
R(x) = (price per unit sold) x
Dollars
C(x) = fixed cost + (price per unit produced) x
Break-even point
x (iGizmo units)

21. Finding a break-even point
A company plans to manufacture electronic age wheelchairs. Fixed cost will be $500,000, and the production cost for each wheelchair is $400. The chairs will be sold at $600 apiece.
Write the Cost function C(x).
Write the Revenue function R(x).
Graph the functions.
Determine the beak-even point.

22. Finding a break-even point
Cost function:
C(x) = 500, x
Revenue function:
R(x) = 600x
Graph

23. Finding a break-even point
Break-even Point C(x) = 500, x R(x) = 600x or y = 500, x y = 600x x = 500, x 200x = 500,000 x = 2500

24. Break-even point
x = y = 600x y = 600(2500) = 1,500, Thus, Break-even point: (2500, 1,500,000) I.e., $1,500,000 with 2500 units sold.

25. The profit function P(x) = R(x) – C(x)
Your Turn
The profit function P(x) = R(x) – C(x)
For the preceding case, P(x) = 600x – (500, x) P(x) = 200x – 500,000
Sketch the graph of the profit function.
What is the y-intercept of the function? How do you interpret that value?
What is the slope of the function? How do you interpret that value?
What is x-intercept of the function? How do you interpret that value?