Solving Multi-Step Equations with Variables on One Side 05:00 minutes

Video Transcript

Transcript Solving Multi-Step Equations with Variables on One Side

Meet Kayla and Sam. They are at the fair and are deciding on how to spend their money on the different attractions.

They want to ride all the rides. The price is 8 dollars per ticket per rollercoaster. They definitely also want to ride the ferris wheel together at least once. The price for the ferris wheel is 5 dollars per ticket.

Sam has a sweet tooth and he can't have a visit to the amusement park without cotton candy. So he will sacrifice a ride on the rollercoaster so he has some money for cotton candy. Cotton candy costs $2.

Together, Kayla and Sam have one hundred dollars. They both want to ride the rollercoasters as often as possible. Our question is: How many rollercoasters can they ride before they run out of money?

To determine the number of times each of them can ride the rollercoaster, we can use a linear equation.

Writing the Linear Equation

Let's collect our information and translate this problem into a linear equation:

  • The rollercoaster is 8 dollars per ride.
  • To ride the ferris wheel Kayla and Sam each have to pay 5 dollars.
  • Cotton candy costs 2 dollars.
  • Together they have 100 dollars to spend.
  • Sam buys 1 stick of cotton candy.
  • They both take a ride in the ferris wheel, which is 2 tickets.

We don't know the number of rides they can take on the rollercoaster, so let x be Kaylas number of rides. Since Sam takes one ride less in order to get some cotton candy, his number of rides can be represented by x − 1.

Let's write our equation:

  • 8 Dollars for the roller coaster times x for the number of rides Kayla takes.
  • Add another 8 dollars for the rollarcoaster times (x-1) for Sam's rides.
  • Then, you add 5 dollars twice for the ferris wheel ride for Kayla and Sam
  • Now, you add 2 dollars for the cost of Sam's cotton candy.
  • They have one hundred dollars to spend so you put this on the right side of the equation.

So now you have a linear equation with the variable x on only one side. You can solve this equation for x to find out how often Kayla and Sam can ride the roller coaster.

Solving the Linear Equation

To start solving this linear equation, you need to follow the order of operations and, using the distributive property, multiply both terms inside the parentheses by eight. Now you have: 8x + 8x − 8 + 5 + 5 + 2 = 100.

On the left side of the equation, combine like terms: Combine the x terms and the integer terms together. The result of this step is 16x + 4 = 100.

Now solve the equation by using opposite operations. Remember that you use PEMDAS reversed. 4 is subtracted from both sides of the equation to obtain all integer values on the right side of the equation. The result of this step is 16x = 96.

To solve for x, use opposite operations again: both sides are divided by 16 in order to isolate the x value on one side of the equation. The solution to the equation is x = 6.

Now, let's check this solution. Substitute the value of 6 in for x. We'll simplify by using the order of operations. First look at the parentheses. Then, complete the multiplication.

Last, you can add all values on the left hand side of the equation. The value on the left side and the right side of the equation are equal. So the solution to the equation x equals 6 is the correct value for x.

So, what does this mean? Remember: We said x was for the number of rides Kayla can take on the roller coaster. So she can take 6 rides. Sam goes 1 less time or x − 1. This is 6 − 1, which is 5. Great!

Now they're finally on the roller coaster. But maybe Sam shouldn't have spent those 2 dollars on the cotton candy right before the rides.