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## 8th grade (Illustrative Mathematics)

### Course: 8th grade (Illustrative Mathematics)>Unit 3

Lesson 9: Lesson 12: Solutions to linear equations

# Worked example: solutions to 2-variable equations

How do you check if an ordered pair is a solution to a given equation? You need to plug in the numbers and see what equality results. Watch this video to see a worked example.

## Want to join the conversation?

• At , would the equation y = -2+ 4/3y be equivalent?
• what if you are not given an ordered pair and you have to figure out this equation
• How are you going to create an equation if you are not given at least one ordered pair? The ordered pair could be implied by giving the x or y intercept, but it is still an ordered pair.
• does this cover absolute vale equalities and inequalities
• I saw nothing here about absolute values which would look like "|x|" or |3|=3 or |-3|=3 OR |-9|=9
The best way to think about absolute values is- "No matter the sign (+ or -) the number remains positive, because the absolute value cannot be negative" Hope that helps. Now inequalities, something simple like....... 1<x<3 where "x" is greater then 1 but less then 3.. The interval notation would be (SET BUILDER NOTATION!)----> {x|x>1, x<3} or (INTERVAL NOTATION(WHICH IS INCLUDING A UNION aka "U"))--> (1,x)U(x,3)...... if the sign was a "greater than or equal to, or less than or equal to" then the interval notation would be [1,x]U[x,3].... I hope this helps.
• What is an "ordered" pair?
• A point on the coordinate plane that gives the horizontal distance from the origin (x) and the vertical distance (y) that is in the form (x,y). If you create a table from a linear equation, you get a series of ordered pairs.
• can't you simplify the equation first? like 4x + 3y = 6?
• You sure can - if you think that will help you.
Look what happened in your case: 4x - 1 = 3y + 5 simplifies to 4x - 3y = 6 not 4x + 3y = 6

Keep Studying!
• solve the following systems of linear equations by elimination
12x+5y=23
2x-7y=39
• Add two times the first equation to the second equation to eliminate x:

12x + 5y = 23
2x - 7y = 39

(12x + 5y) + 2(2x - 7y) = 23 + 2 * 39

26x - 18y = 101

So, y = (26x - 101) / -18. Substitute the expression for y back into the first equation to solve for x:

12x + 5y = 23

12x + 5((26x - 101) / -18) = 23

Expand the right side and simplify:

12x + (130x - 505) / -18 = 23

-6x - 505 / -18 = 23

Multiply both sides by -18 to get rid of the fraction:

6x + 505 = -414

Subtract 505 from both sides to isolate x:

6x = -919

Divide both sides by 6 to find x:

x = -153

Finally, substitute the value of x back into the expression for y:

y = (26x - 101) / -18

y = (26 * -153 - 101) / -18

y = 24.

The solution is x = -153 and y = 24.
• So my math book says that I should rewrite the equation first, simplify it, then graph it. how do I rewrite it? I keep trying and I keep getting the wrong answer.
P.s. It gives me stuff like, 4X + Y = 6
• rewriting for graphing purposes usually means to put in slope intercept form (isolate the y).
Example: 4x + 3y = 9. Subtract 4x to get 3y=-4x+9
Divide all by 3 to get 3/3 y= -4/3 x + 9/3, reduce to get y=-4/3 x + 3. Graph y intercept at (0,3), go down 4 right 3 to get to point (3,-1).
(1 vote)
• I must be missing something that was already said, or something I don't know. But if the equation is like y = -2x - 5. Could you still be able to do what he is doing in the video? I feel like you can but how?
• You can! Anytime you are asked to determine if a point (an ordered pair of (x,y)) is a solution, we use substitution. We use the x-value for "x" in the equation and the y-value for "y" in the equation.

Example: Is (2, 3) a solution to your equation: y = -2x - 5?
Substitute: `3 = -2(2) - 5`
Simplify the right side:
`3 = -4 - 5`
`3 = -9`
Since these are not equal, we know the point (2,3) is not a solution to this equation. Or, is not a point on the line that is created from this equation.

If the 2 sides turn out to be equal, then you know the point is a solution to the equation.
Hope this helps.