Solving Multi-Step Linear Equations

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, $9+6=15$ .

Multi-step linear equations

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 $x–6x+7=52$ .

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

$1x–6x+7=52$

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!

Using the distributive law so solve multi-step linear equations

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 $2\left(4m+5\right)=3m–20$ 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.

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