Question

maliks solution to the equation \\(\frac{2}{5}x - 4y = 10\\), when \\(x = 60\\), is shown below.\\(\frac{2}{5}x - 4y = 10\\)\\(\frac{2}{5}x - 4(60) = 10\\)\\(\frac{2}{5}x - 240 = 10\\)\\(\frac{2}{5}x - 240 + 240 = 10 + 240\\)\\(\frac{5}{2}\left\frac{2}{5}x\
ight = \frac{5}{2}250\\)\\(x = 265\\)what error did malik make first when solving the equation \\(\frac{2}{5}x - 4y = 10\\)?\\(\circ\\) malik did not multiply \\(\frac{5}{2}(250)\\) correctly.\\(\circ\\) malik added 240 to each side of the equation.\\(\circ\\) partially obscured optionsave and exit

Explanation:

Step1: Analyze the substitution step

The original equation is $\frac{2}{5}x - 4y = 10$. When substituting $x = 60$, the correct substitution should be for $x$, but Malik substituted $60$ for $y$ (since the term is $-4y$, he wrote $-4(60)$ which is incorrect as $x = 60$, so the $x$ term is $\frac{2}{5}(60)$, not leaving $x$ as a variable. Wait, actually, looking at the steps: the first error is in the substitution. Wait, the equation is $\frac{2}{5}x - 4y = 10$, and we are to find $y$ when $x = 60$? Wait, no, Malik's steps: he started with $\frac{2}{5}x - 4y = 10$, then wrote $\frac{2}{5}x - 4(60)=10$ – that's the first error. He should substitute $x = 60$ into the $x$ term, not the $y$ term. Wait, but the options: let's check the options. Wait, maybe the original problem is to solve for $y$ when $x = 60$, but Malik treated it as solving for $x$. Wait, the first step: the equation is $\frac{2}{5}x - 4y = 10$, $x = 60$. So substitute $x = 60$: $\frac{2}{5}(60) - 4y = 10$. But Malik wrote $\frac{2}{5}x - 4(60)=10$, which is substituting $y = 60$ instead of $x = 60$ in the $x$ term? Wait, no, he left $x$ as a variable and substituted $60$ for $y$. That's the first error. But among the options (even though some are cut off), the first error is in the substitution. Wait, the first step of Malik's solution: he replaced $y$ with $60$ (since $-4y$ became $-4(60)$) when $x = 60$, so he misassigned the value to the wrong variable. But looking at the given options, maybe the first error is in the substitution step. Wait, the first step where he wrote $\frac{2}{5}x - 4(60)=10$ – that's the first mistake. So the error is in the substitution: he should have substituted $x = 60$ into the $x$ term, not the $y$ term. But among the visible options, maybe the correct option (even though some are cut off) – but let's re-express. Wait, the first error is in the first step of substitution: he substituted $60$ for $y$ (in $-4y$) instead of using $x = 60$ for the $x$ term. So the first error is in the substitution, which is a miscalculation in identifying which variable to substitute. But since the options are partially visible, but the first error is in the substitution step. However, looking at the steps, the first step where he writes $\frac{2}{5}x - 4(60)=10$ is incorrect because $x = 60$, so the $x$ term should be $\frac{2}{5}(60)$, not leaving $x$ as a variable and substituting $60$ for $y$. So the first error is in the substitution, which is a mistake in the variable assignment.

Step2: Evaluate the options

• Option 1: "Malik did not multiply $\frac{5}{2}(250)$ correctly" – this is a later step, not the first error.
• Option 2: "Malik added 240 to each side" – adding 240 is a valid operation (addition property of equality), so that's not an error.
• The third option (cut off) – but the first error is in the substitution step, where he substituted $y = 60$ instead of using $x = 60$ in the $x$ term. So the first error is in the substitution, which is a miscalculation in the variable substitution.

But since the problem is about the first error, and the first step of Malik's solution is $\frac{2}{5}x - 4(60)=10$ (when $x = 60$), the error is that he substituted the value of $x$ (60) into the $y$ term instead of the $x$ term. So the first error is in the substitution, which is a mistake in the variable assignment.

Answer:

(Assuming the correct option is the one about mis-substitution, but since the options are partially visible, but based on the steps, the first error is in the substitution step where he replaced $y$ with 60 instead of using $x = 60$ in the $x$ term. If we consider the given options, the first error is in the first step of substitution, so the answer would be the option related to mis-substitution, but since some options are cut off, but from the visible, the first error is in the substitution: he should have substituted $x = 60$ into $\frac{2}{5}x$ instead of leaving $x$ as a variable and substituting 60 into $y$.)

Wait, maybe the original problem is to solve for $y$ when $x = 60$, so the equation is $\frac{2}{5}(60) - 4y = 10$, but Malik tried to solve for $x$ by substituting $y = 60$, which is wrong. So the first error is in the substitution: he assigned the value 60 to the wrong variable (y instead of using it for x). So the first error is in the substitution step, which is the first step of his solution.