Question

Graph the line that contains the point P and has slope \(\mathrm{m}\). $$ P=(2,4) ; m=-\frac{3}{4} $$

Short answer

The equation of the line is \( y = -\frac{3}{4}x + \frac{11}{2} \).

Step-by-step solution

Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation: \[ y = mx + b \] where \(m\) is the slope and \(b\) is the y-intercept. We will use this form to write the equation of the line.

Identify the Given Information

We are given a point \(P = (2, 4)\) and a slope \(m = -\frac{3}{4}\). Use these values to find the y-intercept \(b\).

Substitute the Point and Slope into the Slope-Intercept Equation

Substitute the point \((2,4)\) into the slope-intercept form to find \(b\): \[ 4 = -\frac{3}{4}(2) + b \]

Solve for the Y-Intercept \(b\)

Multiply out the slope and point values: \[ 4 = -\frac{3}{2} + b \] Add \(\frac{3}{2}\) to both sides to solve for \(b\): \[ 4 + \frac{3}{2} = b \] Convert 4 to a fraction: \[ \frac{8}{2} + \frac{3}{2} = b \] Add the fractions: \[ b = \frac{11}{2} \]

Write the Equation of the Line

Now use the slope \(m = -\frac{3}{4}\) and y-intercept \(b = \frac{11}{2}\) to write the equation of the line: \[ y = -\frac{3}{4}x + \frac{11}{2} \]

Graph the Line

To graph the line, start at the y-intercept \( \left(0, \frac{11}{2}\right) \). From this point, use the slope to find the next point. Since the slope is \(-\frac{3}{4}\), move down 3 units and 4 units to the right from the y-intercept. Plot these points and draw the line through them.

Key concepts

Slope-Intercept Form

To understand how to graph lines, first, we need to talk about the slope-intercept form. This form is a way to write the equation of a line. It's given by \( y = mx + b \). Here, \( m \) represents the slope of the line. The slope tells us how steep the line is and in which direction it goes. The \( b \) term is called the y-intercept. This is the point where the line crosses the y-axis. We use this form because it makes graphing easier. By knowing the slope and y-intercept, we can easily draw the line on a graph.

For example, if you have an equation like \( y = 2x + 3 \), the slope \( m = 2 \) and the y-intercept \( b = 3 \). This means the line crosses the y-axis at 3 and for every 1 unit you go to the right, the line goes up by 2 units.

Finding the Y-Intercept

When graphing a line, finding the y-intercept \( b \) is very crucial. It’s the point where the line crosses the y-axis, which tells us where to start our line on the graph.

To find \( b \), you can use a point \( (x_1, y_1) \) that lies on the line along with the slope \( m \). Using the slope-intercept form \( y = mx + b \), substitute the given point and slope into the equation and solve for \( b \).

Let's break down the steps:

• Start with the slope-intercept equation: \( y = mx + b \).
• Substitute the given point \( (2, 4) \) and slope \( m = -\frac{3}{4} \) into the equation: \( 4 = -\frac{3}{4}(2) + b \).
• Simplify the multiplication \( -\frac{3}{4} \times 2 = -\frac{3}{2} \).
• Add \( \frac{3}{2} \) to both sides: \( 4 + \frac{3}{2} = b \).
• Convert to a common fraction to get \( b = \frac{11}{2} \).

Now we know that the y-intercept \( b = \frac{11}{2} \).

Linear Equations

A linear equation is an equation that forms a straight line when graphed on a coordinate plane. These equations can be written in different forms, but the most common one is the slope-intercept form \( y = mx + b \).

Linear equations show a constant rate of change. This means that the slope (\( m \)) remains the same throughout the entire line. The equation \( y = mx + b \) will always create a line and its slope tells you how steep that line will be.

Let's illustrate this with the given exercise:

• Starting from the point \( P = (2, 4) \) and using the slope \( m = -\frac{3}{4} \), we found the y-intercept \( b = \frac{11}{2} \).
• Using these values, the linear equation for the line is \( y = -\frac{3}{4}x + \frac{11}{2} \).

To graph this, you start at \( (0, \frac{11}{2}) \). From there, move down 3 units and to the right 4 units (as per the slope). This gives you another point on the graph. Connect these points with a straight line, and you have your graph for the equation.