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An equation has to hold a number of important qualities, and if it has those we can use the equal sign as in $3+5=8$ . A linear equation is one in which the highest power of x is 1, and where the variable or variables are multiplied by numbers or added to numbers with nothing more complicated than that. There are no exponents, square roots, or other things like that.

The solution to a linear equation is any number that can be plugged in for the variable that makes a true statement. For example, in a simple, one-step linear equation like $9+x=15$ , we can plug in 6 for x to get a true number statement, $9+6=15$ .

Sometimes, we need more than two operations to solve a linear equation. When that is the case, we use inverse operations to undo each operation in reverse order.

**Example 1**

We need more than two operations to solve the equation $x\u20136x+7=52$ .

We know, because of the Identity Property, that $x=1x$ . So we can write the problem as

$1x\u20136x+7=52$

The next step is to combine the like terms.

$-5x+7=52$

Then we'll undo the addition by subtracting 7 from each side.

$-5x+7-7=52-7$

Which, when simplified, equals

$-5x=45$

Next, we undo the multiplication by dividing each side by 5.

$\frac{-5x}{5}=\frac{45}{5}$

When we simplify that, we get

$x=-9$

And we have solved the multi-step linear equation!

Sometimes multi-step linear equations require us to use the distributive law to solve them. Remember that using the distributive law, or property, means that:

$a\left(b+c\right)=ab+ac$

$\left(b+c\right)a=ba+ca$

**Example 2**

First, solve the equation $2\left(4m+5\right)=3m\u201320$ and then check your solution.

First, use the distributive law on the left side of the equation.

$2\left(4m\right)+2\left(5\right)=3m-20$

Simplify by performing the multiplication on the left side.

$8m+10=3m-20$

Next, subtract 3m from each side.

$8m-3m+10=3m-3m-20$

Simplify the answer by performing the subtraction on each side.

$5m+10=-20$

Next, we're going to undo the addition by subtracting 10 from each side.

$5m+10-10=-20-10$

Simplify by performing the subtraction on each side.

$5m=-30$

Now it's time to undo the multiplication. We do that by dividing each side by 5.

$\frac{5m}{5}=\frac{-30}{5}$

Simplify by performing the division on each side.

$m=-6$

Don't forget we're going to check our solution. To do that, we simply plug in the number -6 into the original equation.

$2\left(\left(4\left(-6\right)\right)+5\right)=\left(3\left(-6\right)\right)-20$

Use PEMDAS to perform the calculations.

$2\left(-24+5\right)=-18-20$

$2\left(-19\right)=-18-20$

$-38=-38$

Since -38 and -38 are the same, we have correctly solved the equation.

Word Problems Involving One Step Linear Equations

College Algebra Diagnostic Tests

As the name suggests, there are a lot of steps involved in solving multi-step linear equations. This can trip some students up when they first start working on them. If your student could use some help learning the steps to take and which order to take them in when solving multi-step linear equations, have them meet with a math tutor who understands how students learn best. Your student's tutor can work with them at their pace, taking extra time with concepts that are especially challenging for them and moving more quickly through concepts that they pick up easily. This way, their tutoring time is efficient as well as effective. To learn more about how tutoring can help your student understand solving multi-step linear equations, contact the helpful Educational Directors at Varsity Tutors today.

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