Chapter 2: Linear Equations and Inequalities
2.1C: Solving One-Step Equations
Solving linear equations is just a matter of undoing operations that are being done to the
variable. The task is always to isolate the variable -- get the variable ALONE on one side
of the equal sign.
Remember when solving equations to "keep the equation
balanced" by making the same changes to BOTH sides of the
equal sign.
Properties of Equality
An equation can be changed into an equivalent equation by:



adding the same quantity to, or subtracting the same quantity from, both sides.
multiplying or dividing both sides by the same nonzero quantity.
Interchanging the two sides, as when 4=x is rewritten as x=4.
Isolating a Variable in One Step
An equation in which the variable is involved in only one arithmetic operation can be
solved in one step by performing the inverse of that arithmetic operation on both sides of
the equation. Inverse operations are operations that undo each other. Addition and
subtraction are inverse operations as are multiplication and division. Performing the
same inverse operation on both sides of the equation insures that the equality or
“balance” of the two sides is not disturbed.
Example 1: In a simple equation, you may only have to undo one operation to solve the
equation.
Solve this equation for x:
x+3=8
The variable is x and we need to get it alone.
In the problem, 3 is being added to the variable,
so to get rid of the added 3, we do the opposite ---x+3=8
subtract 3.
-3 -3
We are actually employing the additive inverse
x =5
property to create a 0 since +3 - 3 = 0. Then the
additive identity is used to get x alone since x + 0 =
x.
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Chapter 2: Linear Equations and Inequalities
2.1C: Solving One-Step Equations
(Remember to subtract 3 from both sides of
the equation to
"keep the equation
balanced".)
Check your answer:
You will always know if your answer is correct by
doing a simple "check" -- substitute your answer
into the original equation and see if the result is
true.
Solve: x  7  5
Solve: x  8  3
Solve:
p
 1.5
4
Solve:  3y  21
Check:
x+3=8
5+3=8
8 = 8 true
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Chapter 2: Linear Equations and Inequalities
2.1C: Solving One-Step Equations
Find the solution set for: b  1.7  5.9
{4.2}
Solve for y:
3
y  21
2
If n  0.17 , express n as the quotient of two integers.
100n=17.17171717
-n=0.1717171
99n=17
n=
17
99
Transposing Terms of an Equation
Solving the equation x+8=3 requires subtracting 8 from both sides, which changes the
original equation into x=3-8. Solving the equation x-7=-5 requires adding 7 to both
sides, which produces the equivalent equation x=-5+7. In each case, the number on the
left side of the original equation was moved or transposed to the opposite side simply by
changing its sign. Here are a few more examples:
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Chapter 2: Linear Equations and Inequalities
2.1C: Solving One-Step Equations
b  1.7  5.9
y  6  1
3  x 1
Checking Possible Solutions
A number satisfies an equation if the equation is true when the variable is replaced by
that number. To verify a number is an actual solution or root of an equation, check that it
satisfies the original equation.
Determine if x=-3 is a root (solution) of the equation 2x  13  7
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Chapter 2: Linear Equations and Inequalities
2.1C: Solving One-Step Equations
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