Writing Linear Equations - Math Steps, Examples & Questions

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Writing linear equations

Here you will learn about writing linear equations, including what they are and how to solve them.

Students will first learn about writing linear equations as part of expressions and equations in $7$ th grade.

What is writing linear equations?

Writing linear equations is when a mathematical situation can be described using algebraic expressions that can be simplified to the form $px+q=r$ (slope-intercept equation) and $p(x+q)=r.$ You can use the algebraic expressions and information about the situation to form an equation and then solve the equation to find the solution.

For example,

Here is a rectangle. It has sides $3x+1$ and $2x-3.$

Form a linear equation for the perimeter, $P,$ in terms of $x.$

The perimeter is the sum of all the sides of the rectangle.

\begin{aligned}P&=2(3x+1)+2(2x-3) \\\\ &=6x+2+4x-6 \\\\ &=10x-4 \end{aligned}

Find the value of $x$ when $P=36.$

\begin{aligned}P&=10x-4 \\\\ 36&=10x-4 \end{aligned}

Solve the equation to find the value of $x,$ by using inverse operations.

So, $x=4.$

Other equations can be written by applying facts from topics, such as:

Angles
Area
Perimeter
Volume
Surface Area
Compound Measures (such as speed, pressure or density)
Sequences
Probability

What is writing linear equations?

Common Core State Standards

How does this relate to $7$ th grade math?

Grade 7: Expressions and Equations (7.EE.B.4a)
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. For example, the perimeter of a rectangle is $54 \, cm.$ Its length is $6 \, cm.$ What is its width?

$[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)$

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

$[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)$

Use this quiz to check your grade 6 to 8 students’ understanding of solving math equations. 10+ questions with answers covering a range of 6th, 7th and 8th grade math equations topics to identify areas of strength and support!

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$[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)$

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

$[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)$

DOWNLOAD FREE

How to form and solve equations

In order to form and solve equations:

Represent the situation algebraically.
Use the information to write the linear equation.
Solve the equation.

Writing linear equations examples

Example 1: ages

Abi is $x$ years old. Rayan is twice as old as Abi. Cam is $3$ years older than Rayan. The total of their ages is $58.$ Form an equation and solve to find $x.$

Represent the situation algebraically.

The situation has three ages. You know that Abi is $x$ years old and need to write expressions for the ages of Rayan and Cam.

Abi is $x$ years old.

Rayan is twice as old as Abi. Therefore Rayan is $2x$ years old.

Cam is $3$ years older than Rayan therefore Cam is $(2x+3)$ years old.

You are given information about the total of their ages, so add the ages together.

\begin{aligned}&x+2x+(2x+3) \\\\ &=5x+3 \end{aligned}

2Use the information to write the linear equation.

The total of the ages is $58.$ So, as $5x+3$ and $58$ both represent the total ages, they are equal to each other.

58=5x+3

3Solve the equation.

First subtract $3$ from both sides of the equation.

Then divide both sides of the equation by $5.$

Check the answer by substituting it back into the original situation.

Abi is $x$ years old and so Abi is $11$ years old.

Rayan is twice as old as Abi and so Rayan is $22$ years old.

Cam is $3$ years older than Rayan and so Cam is $25$ years old.

11+22+25=58

Example 2: perimeter

This is a regular pentagon.

Each side is $3x-4.$

The perimeter of the pentagon is $70.$

Form an equation and solve to find $x.$

Represent the situation algebraically.

The situation is about the perimeter of a regular pentagon. Since it is regular, all the sides of the shape are the same. The perimeter, $P,$ is,

$P=5(3x-4)=15x-20$

Use the information to write the linear equation.

The perimeter is $70$ so, $15x-20$ and $70$ both represent the perimeter of the pentagon, they are equal to each other.

$70=15x-20$

Solve the equation.

First add $20$ to both sides of the equation.

Then divide both sides of the equation by $15.$

Example 3:

Kehlani has walked $102.5$ miles this year. Each week, she walks $7.5$ miles. It will take $x$ weeks for Kehlani to walk $200$ miles in total. Form an equation and solve to find $x.$

Represent the situation algebraically.

The $102.5$ miles is a one-time distance that Kehlani has already walked. The $7.5$ miles is a repeated distance that Kehlani walks each week. Since the number of weeks are represented by $x,$ the additional miles walked each week can be represented by $7.5x.$ Using $T$ to represent the total distance,

$T=102.5+7.5 x$

Use the information to write the linear equation.

The total miles Kehlani wants to walk is $200.$ So, you want to know when $102.5+7.5x$ is the same as $200,$ make them equal to each other.

$200=102.5+7.5x$

Solve the equation.

First subtract $102.5$ from both sides of the equation.

Then divide both sides of the equation by $7.5.$

Example 4: parallel and perpendicular lines

Lines $AB$ and $CD$ are parallel. The line $EF$ intersects lines $AB$ and $CD$ at the points $G$ and $H$ respectively.

Form an equation and solve to find $x.$

Represent the situation algebraically.

The situation is about the parallel lines and angles. The two given angles, $6x-20$ and $4x+15.$

Use the information to write the linear equation.

As alternate angles are equal, the two angles given are equal to each other.

$6x-20=4x+15.$

Solve the equation.

First isolate $x$ to one side of the equation, by subtracting $4x$ from both sides of the equation. Then add $20$ to both sides. Finally divide both sides by $2.$

Example 5: probability

In a box there are only blue counters, green counters, red counters and yellow counters. Dana is going to take at random a counter from the box.

The probability of taking a red counter is twice the probability of taking a yellow counter.

Form an equation and solve to find the probability that Dana picks a yellow counter.

Represent the situation algebraically.

The situation is about the probability of picking counters. You have not been given the probabilities for picking a red counter or a yellow counter, but you have some information.

Let the probability of picking a yellow counter be $x.$ Then the probability of picking a red counter will be $2x.$

Use the information to write the linear equation.

These events are mutually exclusive, so use the fact that the sum of probabilities is $1$ to write the equation.

$0.15+0.25+2x+x=1$

The left hand side of the equation can be simplified by combining the like terms.

$0.4+3x=1$

Solve the equation.

First subtract $0.4$ from both sides of the equation.

Then divide both sides of the equation by $3.$

The question asks for the probability of picking a yellow counter, which is $2x.$

Therefore, the probability of picking a yellow counter is $2\times 0.2=0.4.$

The probability of picking a yellow counter is $0.4.$

Example 6: averages

Charlie is $174\mathrm{~cm}$ tall. Harleen is $x\mathrm{~cm}$ shorter than Charlie.

The mean of their heights is $169\mathrm{~cm}.$

Form an equation and use it to find how tall Harleen is.

Represent the situation algebraically.

The situation is about the mean of two heights. Harleen’s height is $174-x\mathrm{~cm}.$ To find the mean, $M,$ we find the total of the values and divide by the number of values.

$M=\cfrac{174+(174-x)}{2}$

The right hand side can be simplified to:

$M=\cfrac{348-x}{2}$

Use the information to write the linear equation.

As the mean is $\cfrac{348-x}{2}$ and the mean is $169,$ they are equal to each other.

$169=\cfrac{348-x}{2}$

Solve the equation.

First multiply both sides of the equation by $2.$

Then add $x$ to both sides.

Finally, subtract $338$ from both sides.

The question asks for Harleen’s height which is $174-x.$ Therefore, Harleen’s height is $174-10=164.$

Harleen’s height is $164\mathrm{~cm}.$

Teaching tips for writing linear equations

While quality worksheets and practice problems have their place, be sure to connect linear equations to the real world. Find ways to make writing and solving them interactive and applicable to students’ lives.

Even though students may not be graphing lines at this stage, you can still make helpful connections to the written linear equations. Let students know that the root of linear is “line” which is why the equations form straight lines on a coordinate grid. Also that the $x$ and $y$ values relate directly back to the location on the $x$ -axis and $y$ -axis.

Easy mistakes to make

Forgetting that answers can be integers, fractions or decimals
Answers to equations can be integers, fractions or decimals. You can always check if your answer is correct by substituting the answer back into the original situation.

Forgetting to simplify linear equations
Once you have written the situation using algebra, the algebraic equation should be simplified. This may mean expanding using the distributive property and/or simplifying by combining like terms or canceling out terms.

Thinking an equation is not linear when the equation given is not in a certain format
Though commonly shown in slope-intercept, point-slope or standard form, a linear equation does not have to be initially shown this way. Any algebraic expression where the variables have $1$ degree (or an exponent of $1$ ).

Confusing different terms for linear equations
Linear equations can also be referred to as linear functions or linear relationships.

Practice writing linear equations questions

4x+2=58,~x=15

4x-2=58,~x=16

4x-2=58,~x=15

5x-2=58,~x=12

The situation has three ages. You know that Raj is $x$ years old and need to write expressions for the ages of Sam and Tina.

Sam is twice as old as Raj, so Sam is $2x$ years old.

Tina is $2$ years younger than Raj, so Tina is $x-2.$

The total $T$ of their ages is $58.$

\begin{aligned}T&=x+2x+(x-2) \\\\ &=4x-2 \end{aligned}

As the total sum of their ages is $4x-2$ and $58,$ they are equal to each other $4x-2=58$

This linear equation can be solved,

Writing linear equations 13 US

So the solution is $x=15.$

6x+42=108,~x=11

6x+42=108,~x=12

5x+7=108,~x=14.6

6x+7=108,~x=16.8

The shape is a regular hexagon. To find its perimeter, $P,$ we can multiply the side length by $6.$

\begin{aligned}P&=6(x+7) \\\\ &=6x+42 \end{aligned}

As the perimeter is $6x+42$ and $108,$ they are equal to each other.

So the equation is $6x+42=108.$

This equation can be solved,

Writing linear equations 15 US

So the solution is $x=11.$

13,289=1,750+345 m,~m=33

13,289=1,750+345 m,~m=34

13,289=345 m-1,750,~m=43

13,289=345 m-1,750,~m=44

The $\$ 1,750$ is a one-time payment. The $\$ 345$ is a repeated payment that Hector makes each month. The number of months is $m$ and the total of the monthly payments can be represented by $345m.$

Use $T$ to represent the Total amount paid. Then,

T=1,750+345m

Hector needs to pay $\$ 13,289$ in total. So, as the total amount is $1750+345m$ and $\$ 13,289,$ they are equal to each other.

13,289=1,750+345m

This equation can be solved,

Writing linear equations 16 US

Hector needs to pay the entire amount, so it will take $34$ months.

5x=38,~x=7.6

7x-2=180,~x=25.4

5x=38,~x=7.6

7x-2=180,~x=26

The two angles given are supplementary and therefore equal to $180^{\circ}.$

Write the expressions and set them equal to $180.$

\begin{aligned}6x-20+x+18&=180 \\\\ 7x-2&=180 \end{aligned}

This equation can be solved,

Writing linear equations 18 US

4x+0.48=1,~x=0.26

2x+0.48=1,~x=0.13

4x+0.48=1,~x=0.13

2x+0.48=1,~x=0.13

This missing probabilities can be represented by $x$ and $3x\text{:}$

Writing linear equations 20 US

The probability of all the possible events adds up to $1.$

Representing the total of probabilities as $T,$

\begin{aligned}T&=x+0.36+3x+0.12 \\\\ &=4x+0.48 \end{aligned}

As the total of probabilities is $4x+0.48$ and $1,$ they are equal.

So the equation is $4x+0.48=1.$

This equation can be solved,

Writing linear equations 21 US

So the solution is $x=0.13.$

\cfrac{4n+2}{3}=22,~n=16

\cfrac{3n+2}{4}=22,~n=15

\cfrac{3n+2}{4}=22,~n=16

\cfrac{4n+2}{3}=22,~n=15

The total of the three bags is,

n+(n+5)+(2n-3)=4n+2.

\begin{aligned}\text{Mean}&=\cfrac{\text{Total}}{3} \\\\ \text{Mean}&=\cfrac{4n+2}{3} \\\\ 22&=\cfrac{4n+2}{3} \end{aligned}

So the equation is $\cfrac{4n+2}{3}=22.$

This equation can be solved,

Writing linear equations 22 US

So the solution is $n=16.$

Writing linear equations FAQs

In what grades or classes do students work with linear equations?

Students first learn about linear equations in $7$ th grade and continue to work with them in $8$ th grade, $9$ th grade (Algebra $1$ ) and beyond.

What is slope-intercept form?

It is a form of linear equations. In this form, the slope $m$ is the coefficient to $x$ , and the $y$ -intercept is the constant.

What is point-slope form?

This is a different form for the equation of the line. This form is useful for a given point or ordered pair and the slope of the line. It is in the form $y-y_{1}=m(x-x_{1}).$

What is the slope formula?

Slope of a line can be found with the formula $\cfrac{\text{Rise}}{\text{Run}}.$

The next lessons are

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