Download Chapter 4 Part 1: Solving Systems SOLVING SYSTEMS OF

Chapter 4 Part 1: Solving Systems
SOLVING SYSTEMS OF EQUATIONS BY GRAPHING
Remember that a linear equation can be graphed using y = mx + b
When 2 or more linear equations (graphed lines) are graphed together, they form a system of
linear equations.
We have 3 types:
One Solution – When the lines only intersect ONCE. The solution is shown as an ordered pair.
No Solution – When the lines NEVER intersect. EX: Parallel Lines
Infinitely Many Solutions – When the lines graphed (or the equations) are the EXACT SAME.
How to figure it out:
(1) Use the equations that you are given and graph each line.
(2) Look to see if the lines intersect each other at any point.
- 1st - IF the lines intersect ONE TIME then the point they intersect at is the one solution.
- 2nd - If the lines are parallel to each other then they will never intersect and there is
no solution.
- 3rd - If the two lines are identical (they have the same slope and y-intercept), they are
touching at every point on the graph and have infinitely many solutions.
EX: Find the solution for the system of linear equations.
One solution:
y = 2x – 3
(2 , 1)
y=x–1
Notice the two lines intersect at (2,1).
This is a solution to the system of equations.
We can check the solution by using one of the equations and plugging 2 in for x and 1 in for 1 to
see if each side of the equation comes out to equal the other side.
Y = 2x-3
-> 1 = 2(2)-3
-> 1 = 4-3
->
1=1
Chapter 4 Part 1: Solving Systems
Y = -2x + 1
Y = -2x – 1
Since these lines are parallel to each other on the graph, they are considered to have no solution.
2x + 4y = 8
Y = -1/2x + 2
Since these lines are identical once solved in slope intercept form, they are considered to have
infinitely many solutions since both lines are touching and will cross though each point on the
graph at the same time.
Chapter 4 Part 1: Solving Systems
SOLVING SYSTEMS OF EQUATIONS FROM A GRAPH
If an equation or inequality shows a number equal to X, the line will go straight up and down
through that number on the X AXIS.
If an equation or inequality shows a number equal to Y, the line will go left to right through that
number on the Y AXIS.
X=3
Y=4
When it is an inequality we have to shade part of the graph.
≥
OR
> means we shade to the RIGHT if it is an X or ABOVE if it is a Y
≤
OR
< means we shade to the LEFT if it is an X or BELOW if it is a Y
≥ and ≤ Use solid lines
X≥3
AND > and < use dotted lines
Y<4
Chapter 4 Part 1: Solving Systems
SOLVING SYSTEMS OF EQUATIONS USING SUBSTITUTION
When you solve a system of equations using substitution, you will be given a value for either X
or for Y (but not both). Your job is to plug in the value for the variable it represents, then finish
by solving the equation for the other variable.
EX:
Y = 3x + 5
So…
; x=3
Plug 3 in for x and solve for y
Y = 3(3) + 5
->
Y=9+5
->
y = 14
Sometimes instead of telling you exactly what the variable’s value is, they will give you two
equations and ask you to solve for the variables this way. In a case like this one of the 2
equations you are given will show that y = the rest of the problem. To solve, plug the rest of
this equation in for the Y in the other equation and solve.
EX:
Y = -4x + 8
Y = 2x + 2
STEPS:
(1) Write an equation containing only 1 variable and solve
Y = -4x + 8 plug in the other equation for y → 2x + 2 = -4x + 8
+4x
+4x
6x + 2 = 8
→ 6x = 6 → 6x = 6
-2 -2
6
6
X=1
(2) Pick one of the equations to solve for the other variable using your value for x
Y=x+7
Plug in your value for X and solve → y = 2(1) + 2
Y=4
(3) Check your answer to see if each side equals the other
4 ? -4(1) + 8
→ 4 = -4 + 8
→ 4=4