Solving Equations Using the Distributive Property
Hello! Welcome to this video about solving equations using the distributive property. As a quick refresher, remember, if you have something like this
\(3(5x+9)\)
the distributive property says that you can multiply this first term, the outer term, by every term inside the parentheses. So, if we did this we would get:
\(3\times 5x+3\times 9\)
\(15x+27\)
So, now that we’ve reviewed that, let’s use the distributive property to solve some equations. Let’s take a look at the equation:
\(2(3x-7)+x=21\)
So our first step is to simplify the distributive part. The 2 right here is going to be distributed to the \(3x\) and to the -7. Notice that it’s not distributed to this \(+x\) over here. That’s because it’s not inside the parentheses. You only multiply the number outside by each term inside the parentheses; the rest of the part of the equation stays the same.
\(6x-14+x=21\)
So now we can combine like terms on the left side. So \(6x\) and \(x\) are like terms.
\(7x-14=21\)
Now we can solve this just like a normal two-step equation. We’ll start by adding 14 to both sides.
\(7x-14+14=21+14\)
\(7x=35\)
And then all we have to do is divide by 7 on both sides.
\(\frac{7x}{7}=\frac{35}{7}\)
\(x=5\)
And that’s our answer!
Let’s look at another problem.
\(5(2x+6)=-2(3x+9)\)
For this one, we have to apply the distributive property to both sides of the equation, so we’re going to start by doing the left side.
\(10x+30=-2(3x+9)\)
Now, our equal sign stays the same and we’re going to apply the distributive property to the right side of the equation.
\(10x+30=-6x-18\)
Now we can solve it like a regular equation. So, we’re going to start by adding \(6x\) to both sides.
\(10x+6x+30=-6x+6x-18\)
Remember, we want to get all of our \(x\)-terms on one side, and all of our constants on the other side. However you do this is fine; if you wanted to subtract \(10x\) from both sides and get the \(x\)‘s on the right side instead of the left side, that’s totally fine, it would work as well. This is the way I’m going to work it out though; we’re going to get the \(x\)‘s on the left side and the constants on the right side.
\(16x+30=-18\)
Now, we’re going to subtract 30 from both sides, to move this over to the right side of the equation.
\(16x+30-30=-18-30\)
\(16x=-48\)
And finally, we divide by 16 on both sides.
\(\frac{16x}{16}=\frac{-48}{16}\)
\(x=-3\)
And that’s our answer!
I hope that this video was helpful. Thanks for watching and happy studying!
Distributive Property Practice Questions
Solve the equation: \(3\left(2x+4\right)-5=4x-9\).
\(x=8\)
\(x=-8\)
\(x=5\)
\(x=-5\)
Using the distributive property, distribute 3 on the left side of the equation.
\(3\left(2x+4\right)-5=4x-9\)
\(6x+12-5=4x-9\)
Now, combine like terms on the left side of the equation.
\(6x+12-5=4x-9\)
\(6x+7=4x-9\)
To get the variable terms on the left side, subtract 4x from both sides of the equation.
\(6x+7-4x=4x-9-4x\)
\(2x+7=-9\)
Now, solve the two-step equation for \(x\). Subtract 7 from both sides.
\(2x+7-7=-9-7\)
\(2x=-16\)
Divide both sides of the equation by 2.
\(\frac{2x}{2}=\frac{-16}{2}\)
\(x=-8\)
Solve the equation: \(-3\left(2x+6\right)+4x=3\left(x+4\right)\).
\(x=-6\)
\(x=6\)
\(x=2\)
\(x=-3\)
Using the distributive property, distribute –3 on the left side and 3 on the right side of the equation.
\(-3\left(2x+6\right)+4x=3\left(x+4\right)\)
\(-6x-18+4x=3x+12\)
Now, combine like terms on the left side of the equation.
\(-2x-18=3x+12\)
To get the variable terms on the left side, subtract 3x from both sides of the equation.
\(-2x-18-3x=3x+12-3x\)
\(-5x-18=12\)
Now, solve the two-step equation for \(x\). Add 18 to both sides.
\(-5x-18+18=12+18\)
\(-5x=30\)
Divide both sides of the equation by –5.
\(\frac{-5x}{-5}=\frac{30}{-5}\)
\(x=-6\)
Solve the equation: \(-2\left(x+5\right)-3=2\left(3x+1\right)+2x\).
\(x=-\frac{15}{8}\)
\(x=-\frac{8}{15}\)
\(x=\frac{2}{3}\)
\(x=-\frac{3}{2}\)
Using the distributive property, distribute –2 on the left side and 2 on the right side of the equation.
\(-2\left(x+5\right)-3=2\left(3x+1\right)+2x\)
\(-2x-10-3=6x+2+2x\)
Now, combine like terms on the left side and right side of the equation.
\(-2x-13=8x+2\)
To get the variable terms on the left side, subtract 8x from both sides of the equation.
\(-2x-13-8x=8x+2-8x\)
\(-10x-13=2\)
Now, solve the two-step equation for \(x\). Add 13 to both sides.
\(-10x-13+13=2+13\)
\(-10x=15\)
Divide both sides of the equation by –10.
\(\frac{-10x}{-10}=\frac{15}{-10}\)
\(x=-\frac{15}{10}\)
Reducing our answer by 5 gives the answer in simplest form.
\(x=-\frac{15\div5}{10\div5}=-\frac{3}{2}\)
Four more than the product of two and the sum of a number and 1 equals the product of three and the difference of the number and 2. If \(x\) is the number, what is the value of \(x\)?
\(x=9\)
\(x=5\)
\(x=12\)
\(x=-12\)
The product of two and the sum of the number \(x\) and 1 can be written by the algebraic expression \(2(x+1)\). Four more than this expression is \(2\left(x+1\right)+4\). The product of three and the difference of \(x\) and 2 can be written by the expression \(3(x-2)\). Since the 2 algebraic expressions are the same, we have the following equation.
\(2\left(x+1\right)+4=3(x-2)\)
To find the number, distribute 2 on the left side and 3 on the right side of the equation first.
\(2\left(x+1\right)+4=3\left(x-2\right)\)
\(2x+2+4=3x-6\)
Combine like terms on the left side of the equation.
\(2x+6=3x-6\)
To get the variable terms on the left side, subtract \(3x\) from both sides of the equation.
\(2x+6-3x=3x-6-3x\)
\(-x+6=-6\)
Now, solve the two-step equation for x. Subtract 6 from both sides.
\(-x+6-6=-6-6\)
\(\frac{-x}{-1}=\frac{-12}{-1}\)
\(x=12\)
So, the value of the number is \(x=12\).
The length of a rectangle is 3 more than the product of 2 and the sum of its width and 4. If the perimeter of the rectangle is 76 feet, what is its width? Hint: The perimeter, \(P\), of a rectangle is \(P=2l+2w\).
18 feet
9 feet
10 feet
6 feet
Let x be the width of the rectangle. Since the length of the rectangle is 3 more than the product of 2 and the sum of its width and 4, we can represent the length by the expression
\(2\left(x+4\right)+3\). Now, substitute \(l=2\left(x+4\right)+3\), \(w=x\), and \(P=76\) into the equation for the perimeter of a rectangle.
\(P=2l+2w\)
\(76=2[\left(2(x+4\right)+3]+2x\)
To find the width, distribute the 2 on the right side of the equation. Then, distribute 4.
\(76=2[\left(2(x+4\right)+3]+2x\)
\(76=4\left(x+4\right)+2\cdot3+2x\)
\(76=4x+16+6+2x\)
Combine like terms on the right side of the equation.
\(76=6x+22\)
Now, solve the two-step equation for \(x\). Subtract 22 from both sides.
\(76-22=6x+22-22\)
\(54=6x\)
Divide both sides of the equation by 6.
\(\frac{54}{6}=\frac{6x}{6}\)
\(9=x\)
So, the width is 9 feet.
