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Solving Equations
Common Core Standard: Use variables to represent quantities in a real-world or mathematical problem, and construct
simple equations and inequalities to solve problems by reasoning about the quantities.
Common Core Standard: Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q,
and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic
solution, identifying the sequence of the operations used in each approach.
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In 6th grade, students learned to solve one-step equations, now students will learn to solve 1 and 2 step equations
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Example: The perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
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To determine the perimeter of a rectangle there are two commonly used formulas
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1: w + w + l + l, where w = width and l = length
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2: 2w + 2 l
For this example, use the second formula (but you could also use the first one and would still get the same
answer)
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2w + 2 l = P
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Now, substitute the information known in for each variable
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2w + 2(6cm) = 54 cm
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2w + 12 cm = 54 cm
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Combine like terms 12 cm and 54 cm are the constants and like terms
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Subtract 12 cm from each side of the equation to keep the equation equal and because
subtraction is the inverse operation of addition
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2w + 12 cm – 12 cm = 54 cm – 12 cm; Note: 12 cm – 12 cm = 0
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2w = 42 cm
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Now, divide both sides of the equation by 2, because the question asks for what the width equals,
right now, we know what 2 times the width equals
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2w ÷ 2 = 42 cm ÷ 2
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w = 24 cm
Real World Example:
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Jalen spent $200 at the mall this weekend. He bought several items. He purchased three shirts, and he also
bought a pair of shoes for $120. How much were each shirt? Assume that all the shirts cost the same
amount.
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Converting this to an equation can be tricky, take it step by step
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We know that Jalen spent $200, so the equation will equal $200
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We know that he spent $120 for one item
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We also know that he bought 3 shirts for an undetermined amount of money.
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What we don’t know is the variable, so the amount the shirts cost is the variable, x
Now, put this information together
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3x- this represents the amount spent on shirts, since he bought 3 of them
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3x + $120 = this represents the total spent, because he bought a pair of shoes and 3 shirts
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3x + $120 = $200 is the equation
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Combine like terms, which are $120 and $200
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To move $120 with $200, subtract $120, since it is the inverse operation
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3x + $120 – $120 = $200 - $120
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3x = $80
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Now, to determine what one shirt equals, we must change the 3 to a 1 to change a number to 1, divide that
number by itself
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So, divide both sides of the equation by 3, so that the equation stays equal
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3x ÷ 3 = $80 ÷ 3
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x = approximately $26.67.
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So each t-shirt costs about $26.67.
Distributive Property Example:
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Solve: -2(n + 6) = 26
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First, use the distributive property to multiply the terms inside the parenthesis by the -2
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-2 ● n + -2 ● 6 = 26
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-2n + -12 = 26
Now, combine like terms the -12 and 26
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To move the -12 with the 26 add 12 to both sides, since -12 + 12 = 0
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-2n + -12 + 12 = 26 + 12
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-2n = 38
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Now, to determine what just one variable equals, divide the -2n by -2, since this will equal 1
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Also divide 38 by -2, so that the equation stays equal
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-2n ÷ -2 = 38 ÷ -2
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n = - 19
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Prove it!
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-2(n + 6) = 26, n = -19
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Substitute the -19 in for n, since they are equal
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-2(-19 + 6) = 26
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-2(-13) = 26
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-2 ● -13 = 26
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26 = 26, it works!!! 