Question

2y + 3 = -\frac{1}{3}(x - 2)

Explanation:

Step1: Expand the right - hand side

We use the distributive property \(a(b - c)=ab - ac\). Here \(a =-\frac{1}{3}\), \(b=x\) and \(c = 2\). So \(-\frac{1}{3}(x - 2)=-\frac{1}{3}x+\frac{2}{3}\). The equation becomes \(2y + 3=-\frac{1}{3}x+\frac{2}{3}\).

Step2: Isolate the term with y

Subtract 3 from both sides of the equation. We know that 3 can be written as \(\frac{9}{3}\). So \(2y=-\frac{1}{3}x+\frac{2}{3}- \frac{9}{3}\). Simplifying the right - hand side, we get \(2y=-\frac{1}{3}x-\frac{7}{3}\).

Step3: Solve for y

Divide both sides of the equation by 2. When we divide a term by 2, we can also multiply by \(\frac{1}{2}\). So \(y=\frac{1}{2}\times(-\frac{1}{3}x-\frac{7}{3})\). Using the distributive property again, \(y =-\frac{1}{6}x-\frac{7}{6}\).

Answer:

If we want to express \(y\) in terms of \(x\), the solution is \(y =-\frac{1}{6}x-\frac{7}{6}\). If we want to write the equation in standard form \(Ax + By=C\) (where \(A\), \(B\), \(C\) are integers and \(A\geq0\)):

Starting from \(2y+3 =-\frac{1}{3}(x - 2)\), multiply both sides by 3 to get rid of the fraction: \(3\times(2y + 3)=-(x - 2)\).

Expand: \(6y+9=-x + 2\).

Rearrange to get \(x+6y=-7\).