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Warm up
Solve for y
1.
3x + 2y = 5
2.
-4x – 2y = 8
3.
-6x + 3y = -15
Write an equation in slope intercept form
4.
m = 4 and y int (0, 3)
5.
m = -3/2 and y int (2, 4)
answers
1.
2.
3.
4.
5.
y = -3/2x + 5/2
y = -2x – 4
y = 2x – 5
y = 4x + 3
y = -3/2x + 7
Key words
System of Equations
Two or more equations
System Solution
A point (x, y) that satisfies both
equations
Part 1: Finding solutions to
Systems of Equations
There are ____
3 types of solutions
for systems of equations.
One Solution
No Solution
Infinite Solutions
Types of Solutions
One Solution: If there is one solution, then the lines
are intersecting. They may or may not
be perpendicular. The slopes will be
opposite reciprocals if perpendicular.
No Solution:
If there is no solution, then the lines
are parallel. The slopes will be the
same, but “b” will be different.
Infinite
Solutions:
If there are infinite solutions, then
the lines are coinciding. Both “m”
and “b” will be the same.
Solving Systems of Equations
There are 3 ways to solve a system of equations.
Graphing: Graph the lines. Where the
lines intersect is the solution.
Substitution: Solve one of the equations
for one of the variables and substitute.
Elimination: Set up the equations and
combine them to eliminate one of the
variables.
Steps for Using Elimination
1)
2)
3)
4)
5)
Write both equations in standard form
(Ax + By = C) so that variables and = line
up
Multiply one or both equations by a
number to make opposite coefficients for
one of the variables.
Add equations together (one variable
should cancel out)
Solve for remaining variable.
Substitute the solution back in to find
other variable.
Example 1:
5x + y = 12
3x – y = 4
8x
= 16
8
8
x=2
5(2) + y = 12
10 + y = 12
y=2
The solution is: (2, 2)
Step 1: Put both equations in
Already Done
standard form.
Step 2: Check for opposite y and –y are
already
coefficients.
opposites
Step 3: Add equations
together
Step 4: Solve for x
Step 5: Substitute 2 in for x to
solve for y (in either equation)
Example 2
3x + 4y = 9
-x – 4y = 7
Answer: (8, -15/4)
When we need to create opposite
coefficients
Example 3
When you add these neither variable drops out
3x + 5y = 10
3x + y = 2
3x + 5y = 10
-1(3x + y) = -1(2)
SO….
We need to change 1 or both equations by
multiplying the equation by a number that will create
opposite coefficients.
3x + 5y = 10 Multiply the bottom
equation by negative one
to eliminate the x
-3x – y = -2
4y = 8
y=2
Now plug (2) in for y.
3x + 2 = 2
X=0
Solution is : (0,2)
4) 2x + 3y = 6
5x – 4y = -8
4(2x + 3y) = 6(4)
3(5x – 4y) = -8(3)
We will need to change both
equations. We will have the y value
drop out.
8x + 12y = 24
15x - 12y = -24
23 x = 0
x=0
Now plug (0) in for x into any of the 4 equations.
2(0) + 3y = 6
3y = 6
y=2
Solution is: (0, 2)
Solving by Substitution
1.
2.
3.
4.
5.
Solve one equation for either x or y
Substitute the expression into the other
equation
Solve for the variable
Substitute the value back in and solve
Check your answer, is it a solution for both
equations?
Remember that a point consists of an “x” value and a
“y” value. You have to find both to find the solution.
Step 1
Solve one equation for x or y
y=x+1
y = -2x - 2
Already done!
Step 2
Substitute that expression into the other
equation
y=x+1
y = -2x - 2
x + 1 = -2x - 2
Step 3
Solve for the other variable
x + 1 = -2x – 2
+2x
+2x
3x + 1 = -2
- 1
-1
3x = -3
3
3
x = -1
Step 4
Substitute the value back in and solve
y = -1+ 1
y=0
Is (-1, 0) a solution? Check to find out.
0 = -1 + 1
0= -2(-1) – 2
Solution (-1, 0)
Try this one
Ex.
y=x+4
y = 3x + 10
Solution (-3, 1)
Substitution & the distributive property
To use substitution you must have an
equation that has been solved for one of
the variables.
Ex. 3x – 2y = 1
3x-2(x+1) =1
y=x+1
3x –2x -2 =1
x -2 = 1
y= 3 + 1
x=3
y=4
Solution: (3, 4)
Your Turn:
Solve the following systems of equations.
4. y = x +1
(2, 3)
y = 2x – 1
5. y = 2x
7x –y = 15
(3, 6)
You try these:

Tell if lines are parallel, perpendicular,
intersecting but not perpendicular, or coinciding
if one solution, infinite solutions, or many
solutions.

1. 3x + y = 2
3x + y = 6
parallel lines, no solution
3. 5x -10y = 20
-x + 2y = -4
Coinciding lines,
Infinite solution
2. -4x + 3y = 0
4x + y = 8
intersecting lines, one solution
4. y = -1/4 x - 5
-4x + y = 12
perpendicular lines,
one solution
Summary: Draw and Fill in the
table below in your notes.
Parallel
Lines
Graph
Slope & b
Ex. of what
system looks
like
Number of
Solutions
Intersecting
Not ┴
Intersecting
Perpendicular
Coinciding
Lines
Practice
Classwork: CW #7
Homework: WS #7