System Of Equations
By: Eric D.
Solving By Graphing
For this equation you must move the x variable to the opposite side.
Your new equations will be
7y=-2x+14
7y=-5x-7
After getting the y to be by it self, you then need to divide everything by the y variables coefficient. After dividing each side, your new equations will be
y=-2/7x+2
y=-5/7x-1
You then will graph these equations, like on the picture bellow this.
Solving By Substitution
x=10-4y
Now that you have x=10-4y, you have to substitute it into the other equation, 3x-5y=13.
3(10-4y)-5y=13
30-12y-5y=13
30-17y=13
Subtract 30 from each side
-17y=-17
Divide each side by -17
y=1
Now that you have y, you have to put it into both equations to get x, make sure x is the same number for both.
3x-5(1)=13
3x-5=13
Add 5 to each side
3x=18
Divide each side by 3
x=6
Now use this x and y for the other equation
1(6)+4(1)=10
6+4=10
10=10
The answer for this set of equations is x=6 and y=1. (6,1)
Solving By Elimination
-x+3y=5
-3x+8y=8
After doing this step you must make one of the variables equal zero, for example, x+-x=0x, or -5x+5x=0x.
For this set of equations I'll multiply the first equation by negative 3.
(-x+3y=5)-3
3x-9y=-15
Now add this to the other equation.
3x-9y=-15
+-3x+8y=8
You will get
-y=-7
This means y=7.
Now you put y=7 back into the original equations.
3(7)=x+5
21=x+5
Subtract 5 from each side
16=x
For the other equation
-3(16)+8(7)=8
-48+56=8
8=8
The answer for this set of equation is x=16 and y=7. (16,7)
When should I use Substitution? When should I use Elimination? When should I use Graphing?
You should use Elimination whenever you have a question that has the x and y variable on the same side of the equation for both equations.
You should use graphing whenever you have the y variable isolated with and the coefficient is one for both variables.
Types of Solutions
There is No Solution which happens when there is not possible answer. On a graph the would never intersect, they would always be parallel.
There is Infinite Solutions which is when the line always intersect. On a graph they would be a single line.
There is One Solution which is when there is a single answer, it intersects at a single point. On a graph it would be two lines intersecting only for one point of the line.
Solving By Writing The System Of Equations
24.50=10g+1q
22=8g+2q
In this equation, I will use elimination to solve it.
(24.50=10g+1q)-2
-49=-20g-2q
+22=8g+2q
-27=-12g
Divde each side by -12
2.25=g
Putting this into the original equations
24.50=10(2.25)+q
24.50=22.50+q
Subtract 22.50 from each side
2=q
Now try this with your other equations.
22=8(2.25)+2(2)
22=18+4
22=22
This means each quart is 2 dollars,and each gallon is 2.25 dollars. (2.25, 2) (g,q)
Solving Word Problems
x+4y=63
x+6y=87
Like the previous question, I will use elimination.
(x+4y=63)-1
-x-4y=-63
+x+6y=87
2y=24
Divide by 2 on each side
y=12
Now use this for the original equations.
x+4(12)=63
x+48=63
Subtract 48 from each side
x=15
Now use this for the other equation
1(15)+6(12)=87
15+72=87
87=87
This means there is a flat fee of 15 dollars, and a charge of 12 dollars per hour. (15,12)