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One common type of expressions called *linear expressions*. These expressions are useful to model different situations in real life. Because of this, it is sometimes necessary to perform mathematical operations with these expressions. This lesson will explore how to identify linear expressions and how to add or subtract them.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

Dominika really enjoys longboard dancing. She is excited to participate in a competition soon. However, she needs some equipment, so she decided to go to the local skate shop to buy some stuff.

Dominika used her receipt to write out what she spent as an algebraic expression. Maybe her math teacher would give her extra credit when she showed it to him!

While preparing her board for practice, Dominika noticed that she forgot to buy a couple of things, so she had to go back to the store. She wrote another expression for her new purchase.

Now she is ready to practice. On her way to the park to practice, Dominika wondered how to combine the expressions to represent the total she spent at the store. Write an expression for her total expenses.

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Discussion

A linear term is an algebraic expression that includes a coefficient multiplied by a variable with an exponent of one. A linear expression is an expression that includes at least one linear term and any constant terms. No other type of terms may be included. The most common form of a linear expression is given below.

In this expression, $a$ and $b$ are real numbers, with $a =0.$ To completely understand the definition of a linear expression, some important concepts will be be broken down. Consider the example linear expression $x−5y+2.$

$x−5y+2$ | ||
---|---|---|

Concept | Explanation | Example |

Term | Parts of an expression separated by a $+$or $−$sign. |
$x,$ $-5y,$ $2$ |

Coefficient | A constant that multiplies a variable. If a coefficient is $1,$ it does not need to be written due to the Identity Property of Multiplication. | $1,$ $-5$ |

Linear Term | A term that contains exactly one variable whose exponent is $1.$ | $x,$ $-5y$ |

Constant Term | A term that contains no variables. It consists only of a number with its corresponding sign. | $2$ |

The following table shows some examples of linear and non-linear expressions.

Linear Expressions | Non-linear Expressions |
---|---|

$3x$ | $5$ |

$-5y+1$ | $2xy−3$ |

$3x−21 y+2$ | $x1 −2$ |

$πx+6y$ | $5x_{2}+x−1$ |

$3x+4−2x+9−x=13 $

The expression $3x+4−2x+9−x$ looks like a linear expression, but the result after simplifying is $13,$ which is a constant and therefore not a linear expression.Pop Quiz

Determine whether the given expression is a linear expression.

Discussion

When adding or subtracting linear expressions, the process is similar to performing those operations on numbers. As an example, consider two linear expressions.
#### Adding

To add two linear expressions, follow the same process as adding two numbers.
Adding two linear expressions can result in either a new linear expression or a constant, depending on whether the linear terms cancel each other out. #### Subtracting

Subtracting linear expressions is similar to adding them. However, when writing the expression of the subtraction, it is important to distribute the negative sign to change each of terms in the subtrahend expression to its additive inverse.
In summary, when adding and subtracting linear expressions, simplify the resulting expression by combining like terms.

$Expression One:Expression Two: 2x+113x−7 $

Next, these expressions will be added and subtracted. $(2x+11)+(3x−7)⇓2x+11+3x−7 $

The result is an expression with two $x-$terms and two constants. Next, pair up these terms into two sets of like terms. Finally, combine each pair of like terms and simplify. $2x+11+3x−7$

CommutativePropAdd

Commutative Property of Addition

$2x+3x+11−7$

AddTerms

Add terms

$5x+11−7$

SubTerms

Subtract terms

$5x+4$

$(2x+11)−(3x−7)⇓2x+11−3x+7 $

Next, group like terms and simplify the expression.
$2x+11−3x+7$

CommutativePropAdd

Commutative Property of Addition

$2x−3x+11+7$

SubTerms

Subtract terms

$-x+11+7$

AddTerms

Add terms

$-x+18$

Example

Dominika met up with her friends over the weekend to practice for the competition.

She documented her practice times using linear expressions.

Day | Practice Time (minutes) |
---|---|

Friday | $4t+32$ |

Saturday | $7t+26$ |

Sunday | $3t−15$ |

a Write an expression for the total practice time on Friday and Saturday.

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b Dominika and her friends practiced more on Saturday than on Sunday. Write the difference between the practice times on these days.

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a Add the expressions for the practice times on Friday and Saturday.

b Subtract the expression representing Sunday's practice time from the expression for Saturday's practice time.

a Add the practice time from each day to write the total practice time for Friday and Saturday.

$Friday’s Practice Time+Saturday’s Practice Time $

The corresponding expressions can be found in the given table. Day | Practice Time (Minutes) |
---|---|

Friday | $4t+32$ |

Saturday | $7t+26$ |

Sunday | $3t−15$ |

$Friday’s Practice Time+Saturday’s Practice Time$

SubstituteExpressions

Substitute expressions

$4t+32+7t+26$

CommutativePropAdd

Commutative Property of Addition

$4t+7t+32+26$

AddTerms

Add terms

$11t+58$

b This time Sunday's practice time will be subtracted from Saturday's.

$Saturday’s Practice Time−Sunday’s Practice Time $

The corresponding expressions can be found in the given table. Day | Practice Time (Minutes) |
---|---|

Friday | $4t+32$ |

Saturday | $7t+26$ |

Sunday | $3t−15$ |

$Saturday’s Practice Time−Sunday’s Practice Time$

SubstituteExpressions

Substitute expressions

$7t+26−(3t−15)$

Distr

Distribute $-1$

$7t+26−3t+15$

CommutativePropAdd

Commutative Property of Addition

$7t−3t+26+15$

SubTerms

Subtract terms

$4t+26+15$

AddTerms

Add terms

$4t+41$

Example

Finally, it is time for the longboard competition!

Each participant's score is given by adding the two best scores from three rounds. After receiving her scores, Dominika wrote her final score as an expression.$Dominika’s Final Score3x+2y+12 $

Dominika is really proud of her score and she thinks that she can win. After seeing her friend Magdalena perform, though, Dominika starts thinking that maybe Magdalena will win. She then wrote down Magdalena's two best scores. Magdalena's Two Best Scores | |
---|---|

$2x+y+7$ | $x+y+6$ |

In the end, Dominika and Magdalena placed in the top two positions of the competition.

a Write a simplified expression for the difference between Dominika's and Magdalena's scores.

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b Who won first place?

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a Subtract the sum of Magdalena's scores from Dominika's score.

b Who has the higher score?

a The difference between Dominika's and Magdalena's final scores can be found by subtracting Magdalena's final score from Dominika's score.

$Dominika’s Final Score−Magdalena’s Final Score $

Magdalena's final score is not given but it can be found by adding her two best scores. Then the difference between the girls' scores can be written as an expression.
$3x+2y+12−(2x+y+7+x+y+6) $

Here, the sum of two linear expressions is subtracted from another linear expression. Remember that the order of operations must be followed — this means that the expressions between parentheses are simplified first by combining the like terms.
$2x+y+7+x+y+6$

CommutativePropAdd

Commutative Property of Addition

$2x+x+y+y+7+6$

AddTerms

Add terms

$3x+2y+13$

$3x+2y+12−(3x+2y+13)$

Distr

Distribute $-1$

$3x+2y+12−3x−2y−13$

CommutativePropAdd

Commutative Property of Addition

$3x−3x+2y−2y+12−13$

SubTerms

Subtract terms

$-1$

b In Part A, it was found that subtracting Magdalena's final score from Dominika's score results in $-1.$

$Dominika’s Final Score−Magdalena’s Final Score⇓-1 $

This means that Magdalena's score is slightly higher than Dominika's. Since the girls won the top two places, Magdalena won the first place.
$First Place:Second Place: MagdalenaDominika $

Dominika did not win first place in the competition, but she is really proud of herself for winning second place and being so close. Now she is motivated to practice and get even better!
Pop Quiz

Add or subtract the given linear expressions.

Closure

Earlier in this lesson, Dominika wrote a couple of expressions to show how much she spent on two different days buying equipment for the competition.

Dominika wanted to combine the expressions to find one that reflected how much money she spent in total. Notice that the expressions she wrote are linear expressions. To add them, like terms should be grouped together and then added. This means that Dominika spent $$8x+9$ in total when buying the equipment. She is really dedicated to longboard dancing!Loading content