Simultaneous equations result in decimal answer?

Question

I have these 2 equations. $$\left\{ \begin{split} y &= 2x -2 \\ 3y &= -2x + 6 \end{split}\right. $$ I have worked it all out, and plotted the graphs and the point they meet is: $(x,y)=(1,1.5)$.

Can it be this?

I just wasn't sure on the $y=1.5$.

Any help much appreciated.

Answer 1

Converting the second to $y=\frac{-2}{3}x+2$ , helps in showing it's got rational coefficients. to solve for intersection set them equal: $$\begin{eqnarray}2x-2=\frac{-2}{3}x+2\\2x=\frac{-2}{3}x+4\\x=\frac{-1}{3}x+2\\\frac{4}{3}x=2\\x=\frac{6}{4}\end{eqnarray}$$

Plugging this into one of our original equations gives $y=\frac{12}{4}-2=3-2=1$ . So, we see they intersect at (1.5,1)

Answer 2

It's really easy to check: $$ 1 = 2\cdot1.5 - 2\\ 3\cdot 1 = -2\cdot 1.5 + 6 $$ Are these two equalities true? In that case, $x = 1.5, y = 1$ is a solution.

Answer 3

If you mean $x=1.5$ and $y=1$, that is correct. It is wrong to expect that the answers will be integers. If the instructor always picks answers that are integers, that is highly problematic because it gives false impressions.