A student solved the following system of equations incorrectly using these steps:

$\begin{array}{r}
2 x+5 y=13 \\
-x+4 y=13 \\
2 x+5 y=13 \\
+-2 x+8 y=13 \\
\hline 13 y=26 \\
y=2
\end{array}$

$\begin{array}{r}
2 x+5(2)=13 \\
2 x+10=13 \\
2 x=3 \\
x=1.5
\end{array}$

What error did the student make?
A. In line 2, the student forgot to multiply the right side of the equation by 2.
B. In line 3, the student should have subtracted the equations instead of adding them.
C. In line 3, the student added the two equations incorrectly.
D. In line 5, the student used an incorrect equation when solving for $x$.

The student made an error when solving a system of equations using the elimination method. The error was that they multiplied the left side of the second equation by 2 but forgot to multiply the right side by 2. This led to an incorrect value for y , and subsequently, an incorrect value for x . The correct answer is: In line 2, the student forgot to multiply the right side of the equation by 2.
Explanation

Analyzing the Problem The student attempts to solve the system of equations:

2 x + 5 y = 13
− x + 4 y = 13
The student's solution involves multiplying the second equation by 2 and adding it to the first equation. We need to check each step to identify the error.

Correct Solution Let's examine the student's steps:

Original equations:
2 x + 5 y = 13
− x + 4 y = 13

Multiply the second equation by 2:
2 ∗ ( − x + 4 y ) = 2 ∗ 13
− 2 x + 8 y = 26

Add the modified second equation to the first equation:
( 2 x + 5 y ) + ( − 2 x + 8 y ) = 13 + 26
13 y = 39

Solve for y :
y = 13 39 ​ = 3

Substitute y = 3 into the first equation:
2 x + 5 ( 3 ) = 13
2 x + 15 = 13
2 x = − 2
x = − 1


So, the correct solution is ( − 1 , 3 ) .

Identifying the Error Now, let's compare the student's steps with the correct steps to identify the error. The student's steps are:

\begin{array}{r} 2 x+5 y=13 \ -x+4 y=13 \ 2 x+5 y=13 \ +-2 x+8 y=13 \ \hline 13 y=26 \ y=2 \end{array}
The student correctly multiplied the left side of the second equation by 2: − 2 x + 8 y . However, the student forgot to multiply the right side of the equation by 2. The correct equation should be − 2 x + 8 y = 26 , but the student wrote − 2 x + 8 y = 13 .

Conclusion Therefore, the error is: In line 2, the student forgot to multiply the right side of the equation by 2.

Examples
When solving systems of equations, like determining the intersection point of two lines, an error in multiplying one side of an equation can lead to a completely wrong solution. This is similar to balancing a chemical equation; if you don't multiply all parts of a compound correctly, you won't get the right ratios, and the equation won't be balanced. Paying attention to every term and factor ensures accurate results, whether in algebra or chemistry.

Answered by GinnyAnswer | 2025-07-08