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### Course: 8th grade (Illustrative Mathematics)>Unit 4

Lesson 12: Lesson 14: Solving more systems

# Systems of equations with substitution: y=-5x+8 & 10x+2y=-2

Learn to solve the system of equations y = -5x + 8 and 10x + 2y = -2 using substitution. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• What should u do if they ask u to verify it without a graph ?
• Put both equations in slope y-intercept form
If both have the same slope and different
y-intercept then these lines are parallel
and the system has no solution
• To extend this concept, can you solve a linear system of equations with 3 unknowns?
x + y + z = 4
x - y + z = 6
-x + y + z = 0

The three equations are planes in Euclidean three-space (https://en.wikipedia.org/wiki/Three-dimensional_space).
If a solution exists, it's the point (x,y,z) at which all three planes intersect.
• x + y + z = 4 (1)
x - y + z = 6 (2)
-x + y + z = 0 (3)
To find the point in which all three planes intersect, we first find a common coordinate within two equations by elimination or substitution. From there, we use a different pair of equations to find a different coordinate, then plug those two coordinates into all three equations to find and confirm the final coordinate (equations are numbered for clarification):
x - y + z = 6 (2)
-x + y + z = 0 (3)
2z = 6 (elimination)
z = 3

x + y + z = 4 (1)
x - y + z = 6 (2)
2x + 2z = 10 (elimination)
x + z = 5
x + 3 = 5 (We know that z = 3, so we simply plug that in.)
x = 2

x + y + z = 4 (1)
x - y + z = 6 (2)
-x + y + z = 0 (3)

2 + y + 3 = 4
2 - y + 3 = 6
-2 + y + 3 = 0

5 + y = 4
-y + 5 = 6
y + 1 = 0

y = -1
-y = 1
y = -1

y = -1
y = -1
y = -1

That means the solution to this system is (2 , -1 , 3).
• How would I be able to solve the variable for x and y I'm the following equation
x+3y=12
x-y=8
• To solve a system like this, you have two possible methods you can use: elimination or substitution. Substitution will be easy here since you don't have coefficients on several of the variables.
1) pick an equation and isolate a variable
x - y = 8 ---add y to both sides
x = y + 8
2) put this expression in place of the x in the other equation
x + 3y = 12
(y + 8) + 3y = 12 -- group like terms
(y + 3y) + 8 = 12 -- add
4y + 8 = 12 -- subtract 8 from both sides
4y = 4 -- divided by 4
y = 1
3) put this back into the equation from step 1 to find x
x = y + 8
x = 1 + 8
x = 9

• Solve the set of linear equations :
3x-4y=1
4x -3y=6
• How would you solve this problem?
4x + 5y = 11
y = 3x - 13
• Once you find x, you will need to substitute back into one of the equations to find y as well.
y = 3(4) - 13 = -1
Solution is x = 4, y = -1 or (4,-1)
• How would I be able to solve the variable x and y on the following equation

4x-3y=12
x+2y=14
• Luzlilvillalobos,
4x-3y=12
x+2y=14
To solve by substitution, you first need to isolate one of the variables in one equation by itself on one side the equation
Let's isolate the x in the second equation.
x+2y=14 Subtract 2y from both sides
x=14-2y
Now you can substitute (14-2y) for the x in the other equation.
4x-3y=12 Put (14-2y) in for the x
4(14-2y) - 3y = 12
Now you can solve for y
And then put that answer in for y in either original equation and solve for x.

I hope that helps make it click for you
• How would solves equations with squares? For example:

xy=12
x^2+y^2=40
• So if two equations have the same slope then they are parallel and will never intersect? Could you therefore refer to the value on the X coefficients as proof without substitution that a system of equations in slope intersect form has no solution? Like a shortcut.
• Yes, if you can write each equation in slope-intercept form and show that the coefficients on x are the same while the constant terms are different, this would prove that the lines never intersect (no solution).

Have a blessed, wonderful day!