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 the expression for x 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
Problem Analysis Let's analyze the problem. We are given a system of two equations with two variables, x and y , where x represents the higher grade and y represents the lower grade. Our goal is to find the value of x , which is the higher grade.
Expressing x in terms of y The given system of equations is:
x − y = 16 8 1 x + 2 1 y = 52
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
Substitution Now, substitute this expression for x into the second equation:
8 1 ( y + 16 ) + 2 1 y = 52
Eliminating Fractions Multiply both sides of the equation by 8 to eliminate the fractions:
8 × ( 8 1 ( y + 16 ) + 2 1 y ) = 8 × 52 y + 16 + 4 y = 416
Combining Like Terms Combine like terms:
5 y + 16 = 416
Isolating the y term Subtract 16 from both sides:
5 y = 416 − 16 5 y = 400
Solving for y Divide both sides by 5 to solve for y :
y = 5 400 y = 80
Solving 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 x = 96
Final Answer Therefore, the higher grade is 96.
Examples
Understanding systems of equations is useful in many real-world scenarios. For example, suppose you are managing a store and need to determine the price of two products. You know that the difference in price between the two products is $16, and that a combination of one-eighth of the price of the more expensive product and one-half of the price of the less expensive product totals $52. By setting up and solving a system of equations, you can determine the exact price of each product, helping you make informed decisions about pricing and inventory.
We solved the equations to find that the higher grade, represented by x, is 96. This was determined by expressing x in terms of y and solving the equations step by step. Thus, the answer is 96.
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