Question

Decide whether the graphs of the two equations are $$ y=4 x+3 ; 2 y-8 x=-3 $$

Short answer

The two equations are not the same; while they possess the same slope, they have different y-intercepts, and hence, their graphs will not be the same.

Step-by-step solution

Simplify the Second Equation

Let's start with the second equation \(2y - 8x = -3\). To simplify this, divide the entire equation by 2 to get the y-term on one side and you'll get \(y - 4x = -1.5\).

Rearrange the equation

In Step 2, arrange the simplified equation in the form of \(y = mx + b\) which is standard form for a linear equation. Doing this, you get \(y = 4x + 1.5\). The slope \(m\) is 4 and the y-intercept \(b\) is 1.5.

Compare the Equations

After simplifying the second equation, compare it with the first one, \(y = 4x + 3\). After comparing, it is clear to see that while the slope \(m\) is the same for both, the y-intercept \(b\) is different, with the first equation having a y-intercept of 3 and the simplified second one having a y-intercept of 1.5.