Question

Rewrite the equation so that x is a function of y. Then use the result to find x when y = -2, -1, 0, and 1. $$6 y-4(x+3)=-2$$

Short answer

The corresponding x-values for y=-2, -1, 0, 1 are -1/2, 1, 5/2, and 4 respectively.

Step-by-step solution

Rearrange the equation to solve for \(x\)

We start by distributing the 4 into the parenthesis, giving the equation: \(6y - 4x - 12 = -2\). Next, we isolate \(x\) by adding 12 to both sides and adding 4x to both sides which gives \(4x = 6y + 10 \). Finally, we divide every term by 4 to solve for \(x\), this gives us \(x = \frac{6y + 10}{4}\) which simplifies to: \(x = \frac{3y}{2} + \frac{5}{2}\).

Substitute the y-values into the equation

We substitute the given \(y\)-values (-2, -1, 0, 1) into the equation we derived in Step 1. When we substitute \(y=-2\), we get \(x = \frac{3(-2)}{2} + \frac{5}{2} = -3 + \frac{5}{2} = -\frac{1}{2}\). For \(y = -1\), we get \(x = \frac{3(-1)}{2} + \frac{5}{2} = -\frac{3}{2} + \frac{5}{2} = 1\). For \(y = 0\), we get \(x = \frac{3(0)}{2} + \frac{5}{2} = \frac{5}{2}\), and for \(y = 1\), we get \(x = \frac{3(1)}{2} + \frac{5}{2} = \frac{8}{2} = 4\).

Summary of the results

For the given \(y\)-values, the corresponding \(x\) values are: for \(y = -2\), \(x = -\frac{1}{2}\); for \(y = -1\), \(x = 1\); for \(y = 0\), \(x = \frac{5}{2}\); and for \(y = 1\), \(x = 4\).