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(Video) Solving Multi-Step Equations

by Friendly Math 101

Here's a video by Friendly Math 101 showing you how to solve equations where you have to use the distributive property.

$\large{\text{Summary}}$

The first equation she looks at is $5(-7+5a)+3a=105$. Before we begin undoing operations, we need to distribute the $5$ into the parentheses terms. Distributing first allows you to more easily combine like terms and group variables. $$\begin{align} 5(-7+5a)+3a&=105\\ {\color{#8925AE}{5\,\times}}\,(-7)+{\color{#8925AE}{5\,\times}}\,5a+3a&=105\\ -35+25a+3a&=105 \end{align}$$ Next we can combine like terms. $$\begin{align} -35+{\color{#8925AE}{25a+3a}}&=105\\ -35+28a&=105 \end{align}$$ Finally, we isolate the variable by undoing the operations. $$\begin{align} -35\,{\color{#8925AE}{+\,35}}+28a&=105\,{\color{#8925AE}{+\,35}}\\ 28a&=140\\ \frac{28a}{\color{#8925AE}{28}}&=\frac{140}{\color{#8925AE}{28}}\\ a&=5 \end{align}$$

Always remember to solve using PEMDAS once you've used the distributive property!

So, your first step to solve or simplify the equation is the first use the distributive property to solve. Then, you'll combine like terms to reduce the number of variables and constants you're dealing with. Finally, you'll move all of the constants to one side of the equation and the variables to the other. Then, solve like normal!

(Video) An Intro to Solving Linear Equations

by PatrickJMT

In this video, Patrick JMT uses the distributive property to solve some more complex linear equations.

$\large{\text{Summary}}$

The distributive property says that if you've ever got a number multiplied by some stuff within parentheses, you can get rid of the parentheses by multiplying that number by each individual term within them. What that means, to use an example from the video, is that: $$3(m-4) = 3(m) - 3(4)$$

In more general terms: $a(b + c) = ab + ac$.

(Video) Solving Linear Equations

by mathman1024

In this video, he looks at one final problem: $$-[6-(7+x)]=10+3(2x-5)$$ Start by distributing the negative sign on the left and the $3$ on the right. $$\begin{align} {\color{#8925AE}{-}}[6-(7+x)]&=10+3(2x-5)\\ -6+(7+x)&=10+{\color{#8925AE}{3}}(2x-5)\\ -6+(7+x)&=10+6x-15\\ -6+7+x&=10+6x-15 \end{align}$$ Next, we combine like terms on the respective sides. $$\begin{align} {\color{#8925AE}{-6+7}}+x&=10+6x-15\\ 1+x&={\color{#8925AE}{10}}+6x\,{\color{#8925AE}{-\,15}}\\ 1+x&=6x-5 \end{align}$$ Finally, isolate the variable. $$\begin{align} 1+x&=6x-5\\ 1+x\,\color{#8925AE}{-\,x}&=6x\,{\color{#8925AE}{-\,x}}-5\\ 1&=5x-5\\ 1\,\color{#8925AE}{+\,5}&=5x-5\,\color{#8925AE}{+\,5}\\ 6&=5x\\ \frac{6}{\color{#8925AE}{5}}&=\frac{5x}{\color{#8925AE}{5}}\\ \frac65&=x \end{align}$$

The Distributive Property

When you see a complex equation with parentheses in it, the PEMDAS part of your brain should remind you that you need to take care of the parentheses (and the stuff inside them) before you can go any further.

PEMDAS is a trick used to remember the Order of Operations—the sequence of steps you need to take to solve an equation. You start with anything that has parentheses (P), then move on to exponents (E), multiplication (M) and division (D), and finally addition (A) and subtraction (S).

The simplest way of managing parentheses is often through the distributive property. When using this approach, you do exactly what it sounds like: you distribute the things outside the parentheses to the terms inside them.

We'll take a look at a few examples below that combine complex equations and the distributive property.

Let's start with this example: $$5(x - 7) - 4 = 8$$

The first thing we need to do is to handle the distribution of the term outside of the parentheses. In this case, we'll be distributing the $5$ to the term inside the parentheses $(x - 7)$. All it means to distribute the $5$ is to multiply it by each part of the $x - 7$ term.

Think of distributing like dragging a term through the parentheses, almost like paint on a paintbrush. We want to drag the $5$ through the $x$ and the $-7$.

Now that we've handled our parentheses, we can simplify by combining like terms. Combining like terms helps to reduce the number of variables and constants in the equation.

$-35, -4$, and $8$ are like terms. So, we can shuffle them around and combine them: $$\begin{align} 5x - 35 - 4 &= 8\\ 5x - 39 &= 8\\ 5x - 39 \,{\color{#8925AE}{+\, 39}} &= 8 \,{\color{#8925AE}{+\, 39}}\\ 5x &= 47 \end{align}$$

Now all you need to do is solve for $x$.

$5x = 47$ is a one-step equation, and since $x$ is being multiplied by something, we can use division to solve it: $$\begin{align} 5x &= 47\\ \frac{5x}{\color{#8925AE}{5}} &= \frac{47}{\color{#8925AE}{5}}\\ x &= \frac{47}{5} \end{align}$$

Now let's try a more complex example: $$-3[-2(x + 6)] - 11 = 4$$

Don't let the brackets and parentheses worry you. Because we have two sets of them, this just means that we'll need to carry out the distributive property twice. **Start with the innermost set of parentheses first. **

Also, be mindful of your negative signs—they are part of the terms you'll be distributing. For instance, we've got a $-2$ outside the inner set of parentheses. When we distribute, we'll be dragging the $-2$ (not just the $2$) through to each of the terms inside.

$$\begin{align} -3[{\color{#8925AE}{-2}}(x + 6)] - 11 &= 4\\ -3[-2x - 12] - 11 &= 4 \end{align}$$

Now, use the distributive property on the remaining set of parentheses. We'll be dragging the $-3$ through the term inside.

$$\begin{align} {\color{#8925AE}{-3}}[-2x - 12] - 11 &= 4\\ 6x +36 - 11 &= 4 \end{align}$$

Note that the $6x$ and $36$ are both positive. This is because the term we distributed through them was $-3$ (not just $3$). So, we multiplied $-3$ by $-2x$, and then $-3$ by $-12$.

Again, be mindful of your negative signs! They are part of the terms you distribute.

Suddenly our monster-y equation looks a lot more solvable, right? All that's left to do now is solve for $x$.

First, we can combine our like terms: $$\begin{align} 6x + 36 - 11 &= 4\\ 6x + 25 &= 4\\ 6x + 25\,\color{#8925AE}{ -\, 25} &= 4 - 25\,\color{#8925AE}{ -\, 25}\\ 6x &= -21 \end{align}$$

Then we can use division to solve the resulting one-step equation: $$\begin{align} 6x &= -21\\ \frac{6x}{\color{#8925AE}{6}} &= \frac{-21}{\color{#8925AE}{6}}\\ x &= \frac{7}{2}\\ x &= 3.5 \end{align}$$ Done!

When solving problems, you may encounter a situation where you'll need to use the distributive property.

We can use the distribute property as normal to be able to solve this problem. We multiply the $x$ through to the individual values within the parentheses.