Solve equations with variables on both sides of the equal sign.
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In elementary school, students often draw pictures of the arithmetic they do. For instance, they might draw the following picture for the problem $${2+3}$$:
In this picture, each square represents a tile. We can do the same thing for algebraic expressions, but we need to be careful about how we represent the unknown. If we assume that an unknown number of tiles, $$x$$, are contained in a bag, we could draw the following picture for $$2x+3$$:
When we have an equation to solve, we assume that the two sides of the equation are equal. We can represent this by showing them level on a balance. For example, the equation $$2x+3=7$$ could be shown as:
When we solve equations, we can add, subtract, multiply or divide both sides of the equation by the same thing in order to maintain the equality. This can be shown in pictures by keeping the balance level. For example, we could solve the equation $$ 2x+3=7$$ using pictures by first removing (subtracting) 3 from each side, and then splitting (dividing) the remaining blocks into two equal groups:
From this picture, we can see that, in order to keep the balance level, each bag must contain 2 tiles, which means that $$x=2$$.
Solving Equations, accessed on Aug. 31, 2017, 1:47 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
The equation below is solved incorrectly three different ways.
For each solution, find and explain the error(s) that occurred. Then find the correct solution for $${ x}$$.
$${12-4y=7y+10}$$
Incorrect Solution #1 | Incorrect Solution #2 | Incorrect Solution #3 |
$${22=3y}$$ $${7\frac{1}{3}=y}$$ |
$${4y=7y-2}$$ $${-3y=-2}$$ $${y=\frac{2}{3}}$$ |
$${4y=7y+22}$$ $${-3y=22}$$ $${y=-7\frac{1}{3}}$$ |
What value for $$x$$ would make the following equation true?
$$20-(3x-9)-2=-(-11x+1)$$
Grade 8 Mathematics > Module 4 > Topic A > Lesson 6 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..
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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.
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Three expressions are shown below.
Expression A: | $$-\left(8-\frac{1}{2}x\right)$$ |
Expression B: | $$3-1.5x$$ |
Expression C: | $$-2+6x-6$$ |
Using the expressions in the chart, find the value of $$x$$ in each situation described.
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