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1. Equations & Inequalities in One Variable
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United States of AmericaTennessee
Integrated Math 1

1.02 Equations with variables on one side

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Lesson

Concept summary

An equation is a mathematical relation statement where two equivalent expressions and values are seperated 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.

Equations, particularly in real-world contexts, are sometimes referred to as constraints as they describe restrictions or limitations of the given situation.

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:

Distibutive property

a(b+c)=ab + ac

Worked examples

Example 1

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

Approach

Consider how the expression was constructed, starting from the variable. Once this is identified, we can solve the equation by applying the inverse operations in the reverse order.

When applying an operation to both sides of an equation we can justify each step by stating the related property of equality.

Solution

\displaystyle \dfrac{x}{2}+3\displaystyle =\displaystyle 5
\displaystyle \dfrac{x}{2}\displaystyle =\displaystyle 2Subtraction property of equality
\displaystyle x\displaystyle =\displaystyle 4Multiplication property of equality

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 3Multiplication property of equality
\displaystyle 7x+7\displaystyle =\displaystyle 3Combine like terms
\displaystyle 7x\displaystyle =\displaystyle -4Subtraction property of equality
\displaystyle x\displaystyle =\displaystyle -\dfrac{4}{7}Division property of equality

Example 3

Solve the following equation: 0.5x+2\left(1.2x+3\right)=11.8

Solution

\displaystyle 0.5x+2\left(1.2x+3\right)\displaystyle =\displaystyle 11.8
\displaystyle 0.5x+2.4x+6\displaystyle =\displaystyle 11.8Distributive property
\displaystyle 2.9x+6\displaystyle =\displaystyle 11.8Combine like terms
\displaystyle 2.9x\displaystyle =\displaystyle 5.8Subtraction property of equality
\displaystyle x\displaystyle =\displaystyle 2Division property of equality

Example 4

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 to represent the constraint 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}Write an equation in terms of f
\displaystyle 29\displaystyle =\displaystyle \dfrac{85+3f}{5}Combine like terms in numerator
\displaystyle 29\displaystyle =\displaystyle \dfrac{3f+85}{5}Commutative property of addition
\displaystyle 145\displaystyle =\displaystyle 3f+85Multiplication property of equality
\displaystyle 60\displaystyle =\displaystyle 3fSubtraction property of equality
\displaystyle 20\displaystyle =\displaystyle fDivision property of equality

Yolanda received \$20 on the fifth night.

Outcomes

M1.N.Q.A.1

Use units as a way to understand real-world problems.*

M1.N.Q.A.1.C

Define and justify appropriate quantities within a context for the purpose of modeling.*

M1.A.CED.A.1

Create equations and inequalities in one variable and use them to solve problems in a real-world context.*

M1.A.REI.A.1

Understand solving equations as a process of reasoning and explain the reasoning. Construct a viable argument to justify a solution method.

M1.A.REI.B.2

Solve linear and absolute value equations and inequalities in one variable.

M1.A.REI.B.2.A

Solve linear equations and inequalities, including compound inequalities, in one variable. Represent solutions algebraically and graphically.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP5

Use appropriate tools strategically.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

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