Jose asks his friends to guess the higher of two grades he received on his math tests. He gives them two hints:
* The difference of the two grades is 16.
* The sum of one-eighth of the higher grade and one-half of the lower grade is 52.
The system that represents his scores is below.
[tex]
\begin{array}{l}
x-y=16 \\
\frac{1}{8} x+\frac{1}{2} y=52
\end{array}
[/tex]
What is the higher grade of Jose's two tests?
A. 48
B. 52
C. 80
D. 96
Answer (2)
Express x in terms of y using the first equation: x = y + 16 .
Substitute this expression into the second equation: 8 1 ( y + 16 ) + 2 1 y = 52 .
Solve for y : y = 80 .
Substitute the value of y back into the equation x = y + 16 to find x : x = 96 . The higher grade is 96 .
Explanation
Analyze the problem Let's analyze the problem. We are given a system of two equations with two variables, x and y , representing Jose's higher and lower test grades, respectively. Our goal is to find the value of x , which represents the higher grade. The equations are:
x − y = 16 8 1 x + 2 1 y = 52
We will solve this system of equations to find the values of x and y .
Express x in terms of y We can solve this system using substitution or elimination. Let's use substitution. From the first equation, we can express x in terms of y :
x = y + 16
Substitute into the second equation Now, substitute this expression for x into the second equation:
8 1 ( y + 16 ) + 2 1 y = 52
Solve for y Next, we solve for y . First, distribute the 8 1 :
8 1 y + 2 + 2 1 y = 52
Combine the y terms:
8 1 y + 8 4 y = 8 5 y
So the equation becomes:
8 5 y + 2 = 52
Subtract 2 from both sides:
8 5 y = 50
Multiply both sides by 5 8 to isolate y :
y = 50 × 5 8 = 10 × 8 = 80
Solve for x Now that we have the value of y , we can find the value of x using the equation x = y + 16 :
x = 80 + 16 = 96
Conclusion Therefore, the higher grade is 96 and the lower grade is 80.
Examples
Understanding systems of equations is crucial in many real-world applications. For instance, consider a scenario where a bakery sells two types of cakes: chocolate and vanilla. The bakery knows that the total number of cakes sold in a day is 50, and the revenue from chocolate cakes is twice that of vanilla cakes. By setting up a system of equations, the bakery can determine exactly how many of each type of cake were sold, helping them manage inventory and optimize production. This same approach can be applied to various fields, such as economics, engineering, and computer science, where multiple variables and constraints need to be considered simultaneously.
Jose's higher grade is 96. We found this by solving a system of equations representing the relationship between his two test scores. The calculations showed that his lower grade is 80, confirmed by the conditions given in the problem.
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