91Ó°ÊÓ

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.

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 \( b = \frac{11}{2} \).

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.