Creating Equations and Solving System of Equations (Math Problem Sample)

Directions:
Create two equations based on the real-world circumstances given in the word problem below, solve the system of equations using both the graphing and elimination approaches, and then explain your findings.
Problem Description
Julia wants to buy some apples and oranges from a grocery store. The store charges $0.60 per apple and $0.80 per orange. Julia wants to buy a total of 15 fruits and spend exactly $9. What is the number of apples and oranges that Julia should buy?
NOTE: You must show all work!
Problem Analysis
1 Write two equations and define the variables.
Equation 20.6x + 0.8y = 900Equation 20.6x + 0.8y = 9571505080Equation 1x + y = 1500Equation 1x + y = 15
Defining the variables:
Let's define the variables as follows:
Let "x" be the number of apples that Julia buys
Let "y" be the number of oranges that Julia buys
Now we can write two equations based on the given information:
Equation 1: The total number of fruits that Julia buys is 15.
x + y = 15
Equation 2: The total cost of the fruits that Julia buys is $9.
0.6x + 0.8y = 9
These two equations form a system of linear equations, which we can solve using the graphing or elimination method. We will now use these equations to solve for the values of x and y.
2 Solve by graphing.
Graph of;
x+ y = 15
0.6x + 0.8y = 9 is as shown below
Solution: (15, 0)
3 Explain the graphing method, stating the mathematical steps that were taken.
To solve the system of linear equations graphically, we first need to plot the two lines corresponding to each equation on the same coordinate plane.
For Equation 1, x + y = 15, we can plot the line by finding its intercepts:
* When x = 0, y = 15 (y-intercept)
* When y = 0, x = 15 (x-intercept)
For Equation 2, 0.6x + 0.8y = 9, we can plot the line by finding its intercepts:
* When x = 0, y = 11.25 (y-intercept)
* When y = 0, x = 15 (x-intercept)
After plotting both lines, we can find the point of intersection, which represents the solution to the system of equations. In this case, the point of intersection is (6, 9), which means Julia should buy 6 apples and 9 oranges.
The graphing method is a visual approach to solving the system of linear equations. It involves plotting the equations on the same coordinate plane and identifying the point where the two lines intersect. The mathematical steps involved include finding the intercepts for each equation and plotting the resulting lines on the coordinate plane, then identifying the point of intersection by visually inspecting the graph.
4 Now solve using the elimination method.
To solve the system of equations using the elimination method, we need to eliminate one of the variables by multiplying one or both equations by a constant so that the coefficients of one variable are the same in both equations, and then subtract one equation from the other to eliminate that variable.In this case, we can eliminate the variable "y" by multiplying Equation 1 by -0.8 and adding it to Equation 2:-0.8(x + y = 15) -0.8x - 0.8y = -12 0.6x + 0.8y = 9-0.2x = -3Now we can solve for "x" by dividing both sides by -0.2:x = -3/-0.2 = 15We can substitute this value of "x" into either Equation 1 or Equation 2 to solve for "y". Let's use Equation 1:x + y = 15 15 + y = 15 y = 000To solve the system of equations using the elimination method,