Question

1. graph the following using the slope and y-intercept. show all your calculations work.

3x + 4y = 12

Explanation:

Step1: Convert to slope - intercept form

The slope - intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. We start with the equation \(3x+4y = 12\).
Subtract \(3x\) from both sides: \(4y=-3x + 12\).
Then divide each term by 4: \(y=\frac{-3x + 12}{4}=\frac{-3}{4}x+3\).

Step2: Identify slope and y - intercept

From the equation \(y =-\frac{3}{4}x + 3\), we can see that the slope \(m=-\frac{3}{4}\) and the \(y\) - intercept \(b = 3\). The \(y\) - intercept means the line crosses the \(y\) - axis at the point \((0,3)\).

Step3: Use the slope to find another point

The slope \(m = \frac{\text{rise}}{\text{run}}=-\frac{3}{4}\). Starting from the point \((0,3)\) (the \(y\) - intercept), since the rise is \(- 3\) and the run is \(4\), we can go down 3 units and then to the right 4 units. So from \((0,3)\), moving down 3 units gives a \(y\) - coordinate of \(3-3 = 0\), and moving right 4 units gives an \(x\) - coordinate of \(0 + 4=4\). So we get the point \((4,0)\). (We could also go up 3 units and left 4 units: from \((0,3)\), up 3 units gives \(y=3 + 3=6\), left 4 units gives \(x=0-4=-4\), so the point \((-4,6)\) is also on the line.)

Step4: Graph the line

Plot the \(y\) - intercept \((0,3)\) and the other point (for example, \((4,0)\)) on the coordinate plane. Then draw a straight line through these two points.

Answer:

The equation in slope - intercept form is \(y =-\frac{3}{4}x + 3\) with slope \(m =-\frac{3}{4}\) and \(y\) - intercept at \((0,3)\). The line can be graphed by plotting \((0,3)\) and another point (e.g., \((4,0)\)) and drawing a line through them.