Question

sketch the graph of each line: $4x - y = -1$

Explanation:

Step1: Rewrite in slope - intercept form

We start with the equation \(4x - y=-1\). We want to solve for \(y\) to get it in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.
Subtract \(4x\) from both sides: \(-y=-4x - 1\).
Multiply both sides by \(- 1\): \(y = 4x+1\).

Step2: Find the y - intercept

The y - intercept \(b\) is the value of \(y\) when \(x = 0\). For the equation \(y=4x + 1\), when \(x = 0\), \(y=4(0)+1=1\). So the line passes through the point \((0,1)\).

Step3: Find another point using the slope

The slope \(m = 4=\frac{4}{1}\), which means for a run of \(1\) (change in \(x\) of \(1\)), the rise is \(4\) (change in \(y\) of \(4\)). Starting from the y - intercept \((0,1)\), if we increase \(x\) by \(1\) (so \(x = 0 + 1=1\)) and increase \(y\) by \(4\) (so \(y=1 + 4 = 5\)), we get the point \((1,5)\).

Step4: Sketch the line

Plot the points \((0,1)\) and \((1,5)\) on the coordinate plane. Then draw a straight line passing through these two points. The line has a slope of \(4\) (it is a steeply rising line) and crosses the y - axis at \((0,1)\).

Answer:

To sketch the graph of \(4x - y=-1\) (or \(y = 4x + 1\)):

1. Plot the y - intercept \((0,1)\).
2. Use the slope \(m = 4\) to find another point (e.g., \((1,5)\) by moving 1 unit right and 4 units up from \((0,1)\)).
3. Draw a straight line through the two points \((0,1)\) and \((1,5)\).