Download 8-1 Solving Systems of Equations

Systems of Equations Study Guide
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Solving Systems of Equations by Graphing/Estimation
**BASIC IDEA – You graph 2 lines and the point at which they intersect is the solution.
Exceptions
If the lines do not intersect (parallel lines) then you will have NO SOLUTION.
(This means that there are not coordinates that would work in both equations.)
If the lines overlap each other then they are the same equation and therefore have
INFINITELY MANY SOLUTIONS.
(This means that a bunch of different pairs of coordinates would work.)
Example 1:
Graph the system of equations and determine the number of solutions.
If there is only one solution, list the coordinates.
4x – 6y = 12
2x + 2y = 6
** Recall that you would need to change both equations
into slope-intercept form to graph them. (You could also
graph using a table or by using the calculator.)
Example 2:
Graph the system of equations and determine the number of solutions.
Y = 2x + 1
2y = 4x + 2
This answer will be INFINITELY MANY SOLUTIONS.
The two equations are overlapping to create one line.
Example 3:
Graph the system of equations and determine the number of solutions.
Y = -x – 3
Y= -x + 3
The answer is NO SOLUTION. Because the lines are parallel, we know
That means they will never meet; therefore, no common solution.
Solving Systems of Equations by Substitution
**BASIC IDEA- You are taking 2 equations with 2 different variables and writing it as 1
equation with 1 variable.
Example 1
Use substitution to solve the following system of equations.
X + 4y = 1
2x – 3y = -9
Step1 –Choose a variable to solve for in either equation. I will choose x from the first
equation because it has a coefficient of 1 (there is no number next to it) and it will be easy
to solve for x.
X= 1- 4y
Step 2- Now I will take 1-4y and substitute it into the other equation. I will “substitute” 1-4y
for x.
2x- 3y = -9
2(1-4y) – 3y = -9
Step 3- Simplify and solve for y.
2 – 8y – 3y = -9
2-11y = -9
-2
-2
-11y = -11
y =1
This means the y-coordinate of the solution is 1. We still need to figure out what the xcoordinate is.
Step 4- Substitute “1” in for y in either equation and solve for x.
X + 4y = 1
Therefore the point of intersection is (-3, 1)
X + 4(1) = 1
X+4=1
X = -3
Example 2: Use substitution to solve the system of equations. If the system does not have
one solution tell whether it has infinitely many solutions or no solution.
Y = 2x + 1
2y = 4x +2
Step 1- Choose a variable to solve for. Look for the easiest way to solve the problem.
Because the first equation is already solved for y I will choose the first equation.
Y = 2x + 1
Step 2- I will substitute 2x + 1 in for Y in the second equation and solve for y.
2y = 4x + 2
2(2x + 1) = 4x + 2
4x + 2 = 4x + 2
2=2
Infinitely Many Solutions!
Because we end up with 2 = 2 it means that we have infinitely many solutions. If we
graphed the lines they would be overlapping. You may have noticed that the 2 equations
were the same because the second one was exactly 2 times the first equation. There is an
infinite combination of coordinates that would make both equations true. (Don’t confuse
infinite solutions with “All solutions.” All solutions would mean that any random numbers
would make it true but not every combination would work. Not every set of numbers works
but there are an endless amount of coordinates that would work.)
Example 3- Use substitution to solve the system of equations. If the system does not have
one solution tell whether it has infinitely many solutions or no solution.
Y = 2x + 1
Y = 2x + 2
(* You may have a prediction about the answer based on the slopes of the equations  )
Step 1 – Choose either equation and solve for a single variable. This is already done for you. I
will choose
Y = 2x + 1
Step 2 – Substitute 2x + 1 in for y in the other equation.
Y = 2x + 2
( 2x + 1) = 2x + 2
1=2
Because this is never true we have NO SOLUTION for our answer. If we graphed these
equations they would be parallel and therefore no point of intersection. You could have
figured this out from the original problem because their slopes were the same.
Solving Systems of Equations by Elimination (Addition & Subtraction)
**Basic Idea-You are subtracting/adding the 2 equations together to “eliminate” one of
the variables.
Ex. 1 2a + 4c = 30
2a + 2c =21.5
Step 1- Decide whether or not you would add or subtract. Remember that you are trying to
eliminate one of the variables completely. In this example I would want to subtract because
then I would eliminate the “a” terms.
-
2a + 4c = 30
2a + 2c =21.5
2c = 8.5
Step 2- Solve for c by dividing by 2 on both sides.
C = 4.25
Step 3-Substitute 4.25 in for C in either equation and solve for a.
2a + 2c = 21.5
2a + 2(4.25) = 21.5
2a + 8.5
= 21.5
2a = 13
a = 6.5
Example 2
3x – 2y = 4
4x + 2y = 10
Step 1- Determine whether you should add or subtract. In this example I will add because
the -2y and + 2y will cancel out.
3x – 2y = 4
+ 4x + 2y = 10
7x
= 14
Step 2 – Solve for x
X=2
Step 3- Substitute 2 in for x in either equation and solve for y.
3x – 2y = 4
3(2) – 2y = 4
6 – 2y = 4
-2y = -2
y=1
TRY:
1. y = 2x + 1
2y = 4x + 2
SUB
2. 3x – 2y = 4
4x + 2y = 10
Elimination
3. 2a + 4c = 30
2a + 2c =21.5
Graphing
4. y = 2x + 1
y = 2x + 2
Elimination
5. x+ 4y = 1
2x – 3y = -9
Substitution