~ Chapter 7 ~
Systems of Equations & Inequalities
Lesson 7-1 Solving Systems by Graphing
Lesson 7-2 Solving Systems Using Substitution
Lesson 7-3 Solving Systems Using Elimination
Lesson 7-4 Applications of Linear Systems
Lesson 7-5 Linear Inequalities
Lesson 7-6 Systems of Linear Inequalities
Chapter Review
Solving Systems by Graphing
Cumulative Review Chap 1-6
Solving Systems by Graphing
Cumulative Review Chap 1-10
Solving Systems by Graphing
Notes
System of linear equations – Two or more linear equations together… One
way to solve a system of linear equations is by… Graphing.
Solving a System of Equations
Step 1: Graph both equations on the same plane.
(Hint: Use the slope and the y-intercept or x- & y-intercepts to graph.)
Step 2: Find the point of intersection
Step 3: Check to see if the point of intersection makes both equations true.
Solve by graphing. Check your solution.
y=x+5
y = -4x
Your turn…
y = -1/2 x + 2
y = -3x - 3
~ Try another one ~
x+y=4
x = -1
Solving Systems by Graphing
Notes
Systems with No Solution
When two lines are parallel, there are no points of intersection; therefore, the
system has NO SOLUTION!
y = -2x + 1
y = -2x – 1
Systems with Infinitely Many Solutions
y = 1/5x + 9
5y = x + 45
Since they are graphs of the same line… There are an infinite number of
solutions.
Solving Systems by Graphing
Homework
Homework – Practice 7-1
#1-28 odd
Solving Systems by Substitution
Practice 7-1
Solving Systems Using Substitution
Notes
Using Substitution
Step 1: Start with one equation.
Step 2: Substitute for y using the other equation.
Step 3: Solve the equation for x.
Step 4: Substitute solution for x and solve for y
Step 5: Your x & y values make the intersection point (x, y).
Step 6: Check your solution.
y = 2x
7x – y = 15
~ Another example~
Your turn…
c = 3d – 27
y = 4x – 8
y = 2x + 10
4d + 10c = 120
Solving Systems Using Substitution
Notes
Using Substitution & the Distributive Property
3y + 2x = 4
-6x + y = -7
Step 1: Solve the equation in which y has a coefficient of 1…
-6x + y = -7
+6x
+6x
y = 6x -7
Step 2: Use the other equation (substitute using the equation from Step 1.)
3y + 2x = 4
3(6x – 7) + 2x = 4
18x – 21 + 2x = 4
20x = 25
x = 1 1/4
Step 3: Solve for the other variable
Substitute 1 ¼ or 1.25 for x
y = 6(1.25) – 7
y = 7.5 -7
y = 0.5
Solution is (1.25, 0.5)
Solving Systems Using Substitution
Notes
Your turn… 6y + 8x = 28
3 = 2x – y
Solution is (2.3, 1.6) or (2 3/10, 1 3/5)
A rectangle is 4 times longer than it is wide. The perimeter of the rectangle
is 30 cm. Find the dimensions of the rectangle.
Let w = width
Let l = length
l = 4w
2l + 2w = 30
Solve for l…
l = 4(3)
l = 12
Use substitution to solve.
2(4w) + 2w = 30
8w + 2w = 30
10w = 30
w=3
Solving Systems Using Substitution
Homework
Homework ~ Practice 7-2 even
Solving Systems Using Elimination
Practice 7-2
Solving Systems Using Elimination
Notes
Adding Equations
Step 1: Eliminate the variable which has a coefficient sum of 0 and solve.
Step 2: Solve for the eliminated variable.
Step 3: Check the solution.
5x – 6y = -32
3x + 6y = 48
8x + 0 = 16
x=2
Solution is (2, 7)
5x – 6y = - 32
5(2) – 6y = - 32
10 – 6y = -32
-6y = -42
y=7
Check
3(2) + 6(7) = 48
6 + 42 = 48
48 = 48
Your turn… 6x – 3y = 3 & -6x + 5y = 3
Solving Systems Using Elimination
Notes
Multiplying One Equation
Step 1: Eliminate one variable.
-2x + 15y = -32
7x – 5y = 17
Step 2: Multiply one equation by a number that will eliminate a variable.
-2x + 15y = -32
3(7x – 5y = 17)
 -2x + 15y = -32
 21x - 15y = 51
19x + 0 = 19
Step 3: Solve for the variable
19x = 19
x=1
Step 4: Solve for the eliminated variable using either original equation.
-2(1) + 15y = -32
Solution (1, -2)
 -2 + 15y = -32
 15y = -30
 y = -2
Solving Systems Using Elimination
Notes
Your turn… 3x – 10y = -25
4x + 40y = 20
Solution (-5, 1)
Multiply Both Equations
Step 1: Eliminate one variable.
4x + 2y = 14
 3(4x + 2y = 14)
 12x + 6y = 42
7x – 3y = -8
 2(7x – 3y = -8)
 14x – 6y = -16
26x + 0 = 26
Step 2: Solve for the variable
26x = 26
Step 3: Solve for the eliminated variable
4(1) + 2y = 14
x=1
2y = 10
Try this one… 15x + 3y = 9
10x + 7y = -4
y=5
Solution (1, 5)
Solving Systems Using Elimination
Homework
Homework – Practice 7-3 odd
Applications of Linear Systems
Practice 7-3
Applications of Linear Systems
Notes
Applications of Linear Systems
Homework
Homework – Practice 7-4
#6-10
Linear Inequalities
Practice 7-4
Linear Inequalities
Notes
Using inequalities to describe regions of a coordinate plane:
x<1
y>x+1
y ≤ - 2x + 4
Steps for graphing inequalities…
(1) First graph the boundary line.
(2) Determine if the boundary line is a dashed or solid line.
(3) Shade above or below the boundary line… (< below or > above)
Graph y ≥ 3x - 1
Rewriting to Graph an Inequality
Graph 3x – 5y ≤ 10
Solve for y… (remember if you divide by a negative, the inequality sign
changes direction) then apply the steps for graphing an inequality.
Graph 6x + 8y ≥ 12
Linear Inequalities
Homework
Homework ~ Practice 7-5 odd
Systems of Linear Inequalities
Practice 7-5
Systems of Linear Inequalities
Practice 7-5
Systems of Linear Inequalities
Practice 7-5
Systems of Linear Inequalities
Notes
Solve by graphing…
x ≥ 3 & y < -2
You can describe each quadrant using inequalities…
Quadrant I?
Quadrant II?
Quadrant III?
Quadrant IV?
Graph a system of Inequalities…
(1) Solve each equation for y…
(2) Graph one inequality and shade.
(3) Graph the second inequality and shade.
(4) The solutions of the system are where the shading overlaps.
(5) Choose a point in the overlapping region and check in each inequality.
Systems of Linear Inequalities
Notes
Graph to find the solution…
y ≥ -x + 2 & 2x + 4y < 4
Writing a System of Inequalities from a Graph
Determine the boundary line for the pink region…
y=x–2
The region shaded is above the dashed line… so
y>x–2
Determine the boundary line for the blue region…
y = -1/3x + 3
The region shaded is below the solid line… so
y ≤ -1/3x + 3
Your turn…
Systems of Linear Inequalities
Practice 7-6
Homework 7-6 odd
Systems of Linear Inequalities
Practice 7-6
Systems of Linear Inequalities
Practice 7-5
Systems of Linear Inequalities
Practice 7-6
~ Chapter 7 ~
Chapter Review
~ Chapter 7 ~
Chapter Review