Topics
1. Equations & Inequalities in One Variable
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United States of AmericaFL
Grade 912

1.02 Equations with variables on one side

Lesson
Practice
Lesson

Concept summary

An equation is a mathematical relation statement where two equivalent expressions and values are separated by an equal sign. The solutions to an equation are the values of the variable(s) that make the equation true. Equivalent equations are equations that have the same solutions.

One particular type of equation is a linear equation.

Linear equation

An equation that contains a variable term with an exponent of 1, and no variable terms with exponents other than 1.

Example:

2x + 3 = 5

Equations are often used to solve mathematical and real world problems. To solve equations we use a variety of inverse operations and the properties of equality.

Inverse operation

Two operations that, when performed on any value in either order, always result in the original value; inverse operations "undo" each other.

Example:

Operation: Multiply by 2

Inverse: Divide by 2

\dfrac{1}{2}\left(2x\right) = x

The following are the properties of equality:

Reflexive property of equality

a=a

Symmetric property of equality

If a=b, then b=a

Transitive property of equality

If a=b and b=c, then a=c

Addition property of equality

If a=b, then a+c=b+c

Subtraction property of equality

If a=b, then a-c=b-c

Multiplication property of equality

If a=b, then ac=bc

Division property of equality

If a=b and c \neq 0, then a \div c= b \div c

Substitution property of equality

If a=b then b may be substituted for a in any expression containing a.

The following is another important property:

Distributive property

a(b+c)=ab + ac

Worked examples

Example 1

Solve the following equation: \dfrac{x}{2}+3=5

Solution

\displaystyle \dfrac{x}{2}+3\displaystyle =\displaystyle 5
\displaystyle \dfrac{x}{2}\displaystyle =\displaystyle 2Subtract 3 from both sides of the equation
\displaystyle x\displaystyle =\displaystyle 4Multiply both sides of the equation by 2

Example 2

Solve the following equation: x+\dfrac{4x+7}{3}=1

Solution

\displaystyle x+\dfrac{4x+7}{3}\displaystyle =\displaystyle 1
\displaystyle 3x+4x+7\displaystyle =\displaystyle 3Multiply the entire equation by 3
\displaystyle 7x+7\displaystyle =\displaystyle 3Combine like terms
\displaystyle 7x\displaystyle =\displaystyle -4Subtract 7 from both sides of the equation
\displaystyle x\displaystyle =\displaystyle -\dfrac{4}{7}Divide both sides of the equation by 7

Example 3

Yolanda works at a restaurant 5 nights a week and receives tips. On the first three nights, the total tips she received was \$32, \$27, and \$26. She earned twice as much in tips on the fourth night compared to the fifth night. The average amount of tips received per night for the week was \$29.

If the amount she received on the fifth night was \$f, determine how much she received that night.

Approach

The average is equal to the sum of values divided by the number of values. We can use this to build an equation in terms of f about the average tips Yolanda received. Then we can solve the equation for f.

Solution

\displaystyle 29\displaystyle =\displaystyle \dfrac{32+27+26+2f+f}{5}Writing an equation in terms of f
\displaystyle 29\displaystyle =\displaystyle \dfrac{3f+85}{5}Combining like terms in numerator
\displaystyle 145\displaystyle =\displaystyle 3f+85Multiply both sides of equation by 5
\displaystyle 60\displaystyle =\displaystyle 3fSubtract 85 from both sides of the equation
\displaystyle 20\displaystyle =\displaystyle fDivide both sides by 3

Yolanda received \$20 on the fifth night.

Outcomes

MA.912.AR.1.1

Identify and interpret parts of an equation or expression that represent a quantity in terms of a mathematical or real-world context, including viewing one or more of its parts as a single entity.

MA.912.AR.2.1

Given a real-world context, write and solve one-variable multi-step linear equations.

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