Problem 90 - Algebra for College Students

Question

Solve each problem. Jessenda is ordering tacos at \(\$ 0.75\) each and burritos at \(\$ 2\) each for a large group. She must spend \(\$ 300 .\) Write an equation for the total cost and graph it. If she orders 200 tacos, then how many burritos must she order?

Step-by-Step Solution

Answer

Jessenda must order 75 burritos if she orders 200 tacos.

Step 1 - Define Variables

Let’s define two variables: Let x = the number of tacos ordered. Let y = the number of burritos ordered.

Step 2 - Write the Equation for Total Cost

Each taco costs \(0.75, so the cost of x tacos is 0.75x. Each burrito costs \)2, so the cost of y burritos is 2y. The total cost equation is: 0.75x + 2y = 300

Step 3 - Graph the Equation

To graph the equation 0.75x + 2y = 300: 1. Find the x-intercept by setting y = 0: 0.75x + 2(0) = 300 x = 300 / 0.75 x = 400 2. Find the y-intercept by setting x = 0: 0.75(0) + 2y = 300 y = 150 3. Plot the intercepts (400, 0) and (0, 150) on a graph and draw a line through them.

Step 4 - Solve for y when x = 200

Replace x with 200 in the equation: 0.75(200) + 2y = 300 150 + 2y = 300 2y = 150 y = 75 Jessenda must order 75 burritos if she orders 200 tacos.

Key Concepts

solving systems of equations

In mathematics, a system of equations consists of two or more equations that share the same set of variables. To solve these equations means to find the values of the variables that make all the equations true at the same time. There are several methods for solving systems of equations:

• Substitution method: Solve one equation for one of the variables and substitute this value into the other equation.
• Elimination method: Add or subtract the equations to eliminate one of the variables, then solve for the other variable.
• Graphing method: Graph each equation on the same set of axes. The solution is the point where the graphs intersect.

In the original exercise, Jessenda's problem is represented by a single linear equation: 0.75x + 2y = 300. When combined with specific conditions (like x = 200), we can solve it to find the value of y.

graphing linear equations

Graphing linear equations is a visual way to find the solutions of a system. A linear equation in two variables can be written in the form Ax + By = C. Here's the basic process for graphing: First, identify the y-intercept and the x-intercept.

• The y-intercept is where the graph crosses the y-axis (x=0).
• The x-intercept is where the graph crosses the x-axis (y=0).

For the equation 0.75x + 2y = 300, the intercepts are found as follows:
When y = 0: 0.75x = 300 → x = 400
When x = 0: 2y = 300 → y = 150
Plot these points (400, 0) and (0, 150) on a graph and draw a line through them. This line represents all the possible combinations of tacos and burritos that Jessenda can order to spend exactly $300.

intercepts

Intercepts are points where a graph crosses the axes. Finding intercepts is a crucial step in graphing linear equations, as they help to draw the graph accurately. Let's explore the two types in our context: The x-intercept is found by setting y to zero and solving for x. It's the point where the line crosses the x-axis. For our equation, the x-intercept calculation is:
0.75x + 2(0) = 300 → 0.75x = 300 → x = 400.
The y-intercept is found by setting x to zero and solving for y. It's the point where the line intersects the y-axis. For our equation, the y-intercept is:
0.75(0) + 2y = 300 → 2y = 300 → y = 150.
Knowing these intercepts, we can plot the points (400, 0) and (0, 150), helping visualize the relationship between the tacos and burritos Jessenda can afford. This graphical representation makes it easier to determine the required quantities of tacos and burritos for other spending amounts, or in this case when x = 200.