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Chapter 7 Study Guide/Notes
7.1 Solve systems by Graphing
Put both equations in slope intercept form. (Solve for y.)
Graph both equations.
The point where the two lines intersect is the solution.
y = 3x – 5
y=- ½x+2
The solution of this SYSTEM OF EQUATIONS
is the point of intersection of (2,1).
Most systems have ONLY 1 solution.
Some systems have NO SOLUTIONS.
A system with no solutions has two parallel lines.
Some systems have INFINITELY MANY SOLUTIONS.
Those systems have two identical equations. (Solve for y to make sure.)
7.2 Solve systems by Substitution
y = 3x – 4
4x + 3y = 1
Solve for one variable. This one is done for you.
y = 3x – 4
Substitute the expression for that variable
In the other equation.
4x + 3y =1
4x +3 (3x – 4) = 1
Solve the equation.
First distribute.
Combine like terms.
Get the variable term alone.
Divide to get the variable alone.
4x + 9x – 12 = 1
13x – 12 = 1
+12
+12
13 x = 13
13
13
x= 1
Put the value of the variable into either equation to find the value of the other variable.
x= 1
x= 1
y = 3x – 4
4x + 3y = 1
y = 3(1) – 4
4(1) + 3y = 1
y= 3–4
4 + 3y = 1
y = -1
-4
-4
3y = -3
3 3
y = -1
The two values you found [x = ____ , y = ______] are the point (x,y) for the solution of the system of
equations. The solution is (1, -1)
If you solve a system using substitution and you get x = x or 0 = 0 or a number equals itself.
Your system has INFINITELY MANY SOLUTIONS.
If you solve a system using substitution and you get two different numbers equal to each other.
Your system has NO SOLUTION.
7.3 Solve systems by Elimination
This method is different. You will add two equations to eliminate one variable.
3x –y = 7
5x + y = 9
8x + 0 = 16
8x = 16
8
8
x=2
Put x into either equation to find y.
3x – y = 7
5x + y = 9
3(2) – y = 7
5(2) + y = 9
6–y=7
10 + y = 9
-6
-10
-6
-10
-y = 1
y = -1
-1 -1
y = -1
The two values you found [x = ____ , y = ______] are the point (x,y) for the solution of the system of
equations. The solution is (2, -1)
Sometimes you will have to modify the equations by multiplying one or both to be able to eliminate a variable.
2x + 3y = 13
3x + 2y = 12
2x +3(3) = 13
2x + 9 = 13
-9 -9
2x = 4
2 2
x=2
Multiply this equation by 3.
Multiply this equation by -2.
6x + 9y = 39
-6x - 4y = -24 Now add them!
0 + 5y = 15
5y = 15
5
5
y = 3 Put y=3 into either equation.
The solution is (2, 3)
If you solve a system using elimination and you get x = x or 0 = 0 or a number equals itself.
Your system has INFINITELY MANY SOLUTIONS.
If you solve a system using elimination, and you get two different numbers equal to each other.
Your system has NO SOLUTION.
7.4 Application of Linear Systems
7.5 Linear Inequalities p. 407 # 23
7.6 Systems of Linear Inequalities