Question

solve $5kx + 6 = 7kx$ for $x$.

a. $x = \frac{2}{k}$

b. $x = \frac{3}{k}$

c. $x = -\frac{3}{k}$

d. $x = -\frac{2}{k}$

Explanation:

Step1: Subtract \(5kx\) from both sides

To isolate the terms with \(x\), we subtract \(5kx\) from both sides of the equation \(5kx + 6 = 7kx\). This gives us \(6=7kx - 5kx\).

Step2: Simplify the right - hand side

Simplify \(7kx-5kx\) using the rule of combining like terms. Since \(7kx - 5kx=(7 - 5)kx = 2kx\), the equation becomes \(6 = 2kx\).

Step3: Solve for \(x\)

Divide both sides of the equation \(6 = 2kx\) by \(2k\) (assuming \(k
eq0\)). We have \(x=\frac{6}{2k}=\frac{3}{k}\)? Wait, no, wait. Wait, let's re - do step 1. Wait, maybe I made a mistake. Let's start over.

Wait, original equation: \(5kx+6 = 7kx\)

Subtract \(5kx\) from both sides: \(5kx+6-5kx=7kx - 5kx\)

Which simplifies to \(6 = 2kx\)

Now, divide both sides by \(2k\): \(x=\frac{6}{2k}=\frac{3}{k}\)? But wait, that's not matching. Wait, no, wait, maybe I messed up the sign. Wait, no, let's check again.

Wait, maybe the correct step is:

Start with \(5kx + 6=7kx\)

Subtract \(7kx\) from both sides: \(5kx+6 - 7kx=7kx-7kx\)

Which is \(- 2kx+6 = 0\)

Then subtract 6 from both sides: \(-2kx=-6\)

Then divide both sides by \(- 2k\): \(x=\frac{-6}{-2k}=\frac{3}{k}\)? No, that's not right. Wait, no, wait, let's do it properly.

Wait, \(5kx+6 = 7kx\)

Subtract \(5kx\) from both sides: \(6=7kx - 5kx\)

\(7kx-5kx = 2kx\), so \(6 = 2kx\)

Then \(x=\frac{6}{2k}=\frac{3}{k}\). But the options have \(x = \frac{3}{k}\) as option B? Wait, but let's check the options again.

Wait, the options are:

A. \(x=\frac{2}{k}\)

B. \(x=\frac{3}{k}\)

C. \(x =-\frac{3}{k}\)

D. \(x=-\frac{2}{k}\)

Wait, maybe I made a mistake in the sign. Let's try subtracting \(7kx\) from both sides:

\(5kx + 6-7kx=7kx - 7kx\)

\(-2kx+6 = 0\)

\(-2kx=-6\)

\(x=\frac{-6}{-2k}=\frac{3}{k}\). So the answer should be B. \(x = \frac{3}{k}\)

Wait, but let's check with an example. Let \(k = 3\). Then the original equation is \(5\times3x+6=7\times3x\)

\(15x + 6=21x\)

Subtract \(15x\) from both sides: \(6 = 6x\)

\(x = 1\)

Now, if \(k = 3\), then \(\frac{3}{k}=\frac{3}{3}=1\), which matches. So the correct answer is B.

Answer:

B. \(x=\frac{3}{k}\)