Standard Form 05:16 minutes

Video Transcript

Transcript Standard Form

Kayla runs a lemonade and chocolate chip cookies stand during summer vacation to make some money.

She has just sold her last cup of lemonade for the day. She made a total of $180 and sold 80 chocolate chip cookies.

Kayla wants to know how many cups of lemonade she has sold to prepare for tomorrow's sales.

Writing the Equation in Standard Form

A linear equation in standard form can be used to find the number of cups of lemonade.

The standard form of a linear equation is Ax + By = C, where A and B cannot both equal zero.

Let's use x to represent the number of cups of lemonade sold and y for the number of cookies sold.

  • We also know that the cost per cup of lemonade is 2 dollars.
  • And that Kyla sells her cookies for 1 dollar and 50 cents each.
  • We also know that Kyla sold a combined hundred eighty dollars' worth of cookies and lemonade.

Our coefficients in this example are the prices we know. So 2 can be substituted in for A. Now we have an expression that represents the money earned from the lemonade: 2x.

$1.50 can be substituted in for B. 1.5y stands for the money earned from the cookies.180 can be substituted in for C. This represents Kyla's sales total for the day. So our expression to find the number of cups of lemonade Kayla sold, written in standard form, is 2x + 1.5y = 180.

Solving the Linear Equation

Since we know Kyla sold 80 cookies, we can plug this into y. Now we can solve the equation for x in order to find the number of cups of lemonade Kyla sold.

  • The first step is to multiply 1.5 and 80 together, giving us 120. Remember, 120 is the total money Kyla earned from selling cookies.
  • The next step is to subtract 120 from both sides of the equation, leaving us with 2x = 60.
  • The last step is to divide both sides of the equation by 2, leaving us with x = 30. So now we know Kayla sold 30 cups of lemonade today.

Graphing Linear Equations in Standard Form

A linear equation in standard form can also be graphed to determine the number of cups of lemonade and cookies sold for 180 dollars. The linear equation produces all possible combinations of lemonade and cookie sales that would yield $180 including negative values.

The part of the graph we need to focus on is in the first quadrant where x and y are positive. The x- and y-intercepts can be used to find the starting point and ending point for the number of items sold.

The x-intercept will show the maximum amount of lemonade that can be sold for $180. The y-intercept will show the maximum amount of cookies that can be sold for $180.

Finding the X-Intercept

The first step is to find is the x-intercept. This is the point that crosses the x-axis when y is 0. This means we should substitute y in the equation with 0.

Since anything times 0 equals 0, this leaves us with the equation 2x = 180. The last step is to divide both sides of the equation by 2, giving us x equals 90. Now we have our x-intercept. And we can graph the ordered pair of (90, 0) on the coordinate plane.

Finding the Y-Intercept

The second step is to find is the y-intercept. This is the point that crosses the y-axis when x is 0. This means we should substitute x in the equation with 0. Since anything times 0 equals 0, this leaves us with the equation 1.5y = 180.

The last step is to divide both side of the equation by 1.5 giving us y equals 120. Now, we have our y-intercept. So we can graph the ordered pair of (0, 120) on the coordinate plane.

Example Solutions

We can use the graph to see how many cups of lemonade sold when she sold 80 cookies. When y equals 80, we look at our line and see that x equals 30 cups of lemonade.

We can use the graph to see the relationship between the number of cups of lemonade and number of cookies sold for Kayla to make $180.

Another example is Kayla would have made $180 by selling 60 cups of lemonade and 40 cookies.

The next day Kayla comes to open her stand. Oh no! All of the neighborhood kids want to get in on her profit!