$\left{\begin{array}{l} 9 x-y=15 \ 2 x+8 y=28 \end{array}\right. Raj's Work $\begin{array}{rlrl} 9 x-y & =15 & 2 x+8(15+9 x) & =28 \ -y & =15-9 x & 2 x+120+72 x & =28 \ y & =-15+9 x & 74 x+120 & =28 \ & 74 x & =-92 \ & x & =-\frac{46}{37} \end{array}$ 1 2 3 4 5 $\begin{aligned} 9\left(-\frac{46}{37}\right)-y & =15 \ -\frac{414}{37}-y & =15 \ y & =-\frac{969}{37} \end{aligned}$ What error did Raj make? A. Raj forgot to multiply the value of $x$ by 9 when solving for the value of $y$. B. Raj subtracted 120 from both sides when he solved for the value of $x$. C. Raj forgot the negative sign when substituting $-15+9 x$ for $y$. D. Raj found the value of $-y$ instead of $y$.
Answer (2)
Isolate y in the first equation: y = 9 x − 15 .
Substitute the expression for y into the second equation: 2 x + 8 ( 9 x − 15 ) = 28 .
Simplify and solve for x : 74 x − 120 = 28 ⟹ 74 x = 148 ⟹ x = 2 .
Substitute the value of x back into the equation for y : y = 9 ( 2 ) − 15 = 3 . The error was in Raj's calculation when solving for x . The correct answer is: Raj subtracted 120 from both sides when he solved for the value of x .
Explanation
Analyzing the Problem We are given a system of equations and Raj's attempt to solve it. Our goal is to identify the error in Raj's work. The system of equations is:
{ 9 x − y = 15 2 x + 8 y = 28
Raj's work involves isolating y in the first equation and substituting it into the second equation.
Isolating y Let's examine Raj's steps. First, he isolates y in the first equation:
9 x − y = 15 − y = 15 − 9 x y = 9 x − 15
Raj has y = − 15 + 9 x , which is equivalent to y = 9 x − 15 . So far, so good.
Substitution Next, Raj substitutes this expression for y into the second equation:
2 x + 8 y = 28 2 x + 8 ( − 15 + 9 x ) = 28
This step is also correct.
Solving for x Now, let's simplify the equation and solve for x :
2 x + 8 ( − 15 + 9 x ) = 28 2 x − 120 + 72 x = 28 74 x − 120 = 28 74 x = 28 + 120 74 x = 148 x = 74 148 x = 2
Raj has 74 x + 120 = 28 , which leads to 74 x = − 92 , and x = − 37 46 . This is where Raj made an error. He subtracted 120 from both sides incorrectly. It should be adding 120 to both sides.
Correcting x and Solving for y Let's correct Raj's mistake and find the correct value for x :
74 x − 120 = 28 74 x = 148 x = 74 148 x = 2
Now, substitute the correct value of x back into the equation for y :
y = 9 x − 15 y = 9 ( 2 ) − 15 y = 18 − 15 y = 3
Identifying the Error The error Raj made was subtracting 120 from both sides when he solved for the value of x . He should have added 120 to both sides, resulting in 74 x = 148 , not 74 x = − 92 .
Final Answer The correct answer is: Raj subtracted 120 from both sides when he solved for the value of x .
Examples
When solving systems of equations, it's crucial to accurately manipulate algebraic expressions to find the correct variable values. For instance, consider a scenario where you're mixing two different types of juice to achieve a specific concentration and volume. Setting up a system of equations helps determine the exact amount of each juice needed. Correctly solving for these amounts ensures the final mixture meets the desired specifications, preventing errors in the recipe or formulation.
Raj's error occurred when he subtracted 120 from both sides of the equation instead of adding it, which led to an incorrect value for x. The corrected process shows that x equals 2 and y equals 3. Therefore, the chosen answer is option B.
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