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Solving a linear equation breaks down to a few key steps:

1. Use the distributive property to get rid of parentheses. The reason we do this is so we can more easily combine like terms in step 2.
2. Use arithmetic properties to simplify each side of the equation. We can do this by combining like terms on each side of the equation.
3. Group like terms so all the unknowns (the variable) are on one side of the equation and the constants are on the other side of the equation.
4. Use arithmetic properties again to simplify to get a value for the unknown.

These steps should give you a solution for any linear equation you encounter.

Just remember: whatever you do on one side of the equation, you have to do on the other in order to keep the equation equivalent!

We can solidify these steps by looking at how they work in an example.

Suppose we want to solve $$2(x-7) + 10 = 2x+7(x-1)\text{ for }x$$

As we see above, we first want to use the distributive property to get rid of the parentheses. $$\begin{align*} {\color{#8925AE}{2}}(x-7) + 10 &= 2x+{\color{#8925AE}{7}}(x-1)\\ 2x - 2\times 7 + 10 &= 2x + 7x - 7\times 1 \end{align*}$$

Now that there are no more parentheses, we can simplify each side individually. $$\begin{align*} 2x - {\color{#8925AE}{2\times 7}} + 10 &= 2x + 7x - {\color{#8925AE}{7\times 1}}\\ 2x - {\color{#8925AE}{14 + 10}} &= {\color{#8925AE}{2x + 7x}} - 7\\ 2x - 4 &= 9x - 7\\ \end{align*}$$

We simplified each side as much as we could, so now we move on to grouping the unknown variable (in this case, this is $x$) on one side and the rest of the terms on the other side.

$$\begin{align*} 2x - 4 &= 9x - 7\\ 2x - 4\, \color{#8925AE}{-\,9x}&= 9x \,{\color{#8925AE}{-\,9x}} - 7\\ -7x - 4&= - 7\\ -7x - 4\, \color{#8925AE}{+\,4}&= - 7\,\color{#8925AE}{+\,4}\\ -7x&= - 3\\ x &= \frac{3}{7} \end{align*}$$

So we got our answer! It is always a good idea to check your answers, so try plugging $x=\frac{3}{7}$ into the original equation to see if it works.

(Video) Solve Linear Equations by Grouping

by PatrickJMT

Patrick JMT has a good video example of a linear equation he solves by grouping.

$\large{\text{Summary}}$

He uses a couple of steps:

1. Get rid of the parentheses using the distributive property.
2. Group like terms to make sure that all the $x$'s are all on the same side, and all the numbers are on the other.
3. Finish solving the equation by using division or multiplication.

Let's work though one of his examples: $$-(-3+x)+4=2(x+3)-6x+7$$

First, we're going to use the distributive property to get rid of parentheses. $$\begin{align} -(-3+x)+4&=2(x+3)-6x+7\\ -1\cdot -3 + -1 \cdot x + 4 &= 2\cdot x + 2\cdot 3 - 6x+7\\ 3-x+4&=2x+6 -6x +7 \end{align}$$

Now, we can combine like terms on either side of the equation. $$\begin{align}3-x+4&=2x+6 -6x +7\\ -x+7&=-4x+13\end{align}$$

We can subtract $7$ and add $4x$ to both sides to isolate the variable on one side. $$\begin{align}-x+7&=-4x+13\\ -x&=-4x+6\\ 3x&=6 \end{align}$$

The final step is to divide both sides by $3$. $$\begin{align} 3x&=6\\ x&=2\end{align}$$

Solve by Grouping

Solving linear equations by grouping is pretty simple once you get the hang of it. All you have to do is use operations (addition, subtraction, multiplication, and division) to put like terms on different sides of the equation. Let's look at an example. $$2x+5x-3-6+9-12=6x+4+1+5+7-2+7x-3$$