Question

Sketch the line through the given point with the indicated slope. $$ (-2,3) ; 4 $$

Short answer

The equation of the line is \(y = 4x + 11\), and we found an additional point \((-1, 7)\) on the line. By plotting \((-2, 3)\) and \((-1, 7)\) and connecting them, we can sketch the line with the indicated slope of 4.

Step-by-step solution

Find the equation of the line

Given point: \((-2, 3)\) Slope: \(4\) Let's recall the point-slope form of a linear equation, which is given by: \[y - y_1 = m(x - x_1)\] where \((x_1, y_1)\) is the given point and \(m\) is the slope. Plugging in the given point \((-2,3)\) and the slope \(4\), we get: \[y - 3 = 4(x - (-2))\]

Simplify the equation

Now, let's simplify the equation further: \begin{align*} y - 3 &= 4(x + 2) \\ y - 3 &= 4x + 8 \\ y &= 4x + 11 \end{align*} This is the equation of the line in slope-intercept form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the \(y\)-intercept.

Identify additional points on the line

Now let's find some additional points on the line to help us sketch it. Since the slope is \(4\), for every \(1\) unit increase in the \(x\)-value, the \(y\)-value will increase by \(4\) units. Starting at the given point \((-2,3)\), if we increase the \(x\)-value by \(1\) unit, we have: \[-2 +1 = -1\] Then, we increase the \(y\)-value by \(4\) units: \[3 + 4 = 7\] So, another point on the line is \((-1,7)\).

Sketch the line

Finally, let's sketch the line using the given point \((-2,3)\), the new point \((-1,7)\), and the equation of the line \(y = 4x + 11\). 1. Plot the given point \((-2,3)\) and the new point \((-1,7)\) on a graph. 2. Draw a straight line through these points, extending the line on both sides. 3. Write the equation of the line \(y = 4x + 11\) next to the line. The sketched line represents the line that goes through the given point with the indicated slope.