Problem 5 - Algebra A Combined Function

Question

Write an equation of each line with the given slope and containing the given point. Write the equation in the slope-intercept form \(y=m x+b .\) See Example \(1 .\) Slope \(\frac{1}{2} ;\) through (-6,2)

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{1}{2}x + 5 \).

Step 1: Understand the Problem

We need to write the equation of a line that has a slope of \( \frac{1}{2} \) and passes through the point \((-6, 2)\). The equation should be in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Identify the Slope

The slope \( m \) is given as \( \frac{1}{2} \). We will use this value directly in our equation.

Step 3: Use the Point Slope Formula

To find the y-intercept \( b \), we use the point slope formula: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the given point. Here, \( (x_1, y_1) = (-6, 2) \). So, \( 2 - y = \frac{1}{2}(x + 6) \).

Step 4: Substitute and Simplify

Plug in \( y = 2 \), \( x = -6 \) and \( m = \frac{1}{2} \) into the point slope formula to solve for \( b \). Simplifying, we have: \[ 2 = \frac{1}{2}(-6) + b \] Calculating further, \[ 2 = -3 + b \] Solving for \( b \), we find \[ b = 5 \].

Step 5: Write the Final Equation

Now we substitute \( m = \frac{1}{2} \) and \( b = 5 \) into the slope-intercept form. Therefore, the equation of the line is \( y = \frac{1}{2}x + 5 \).

Key Concepts

line equation

A line equation is a mathematical expression that describes all the points along a straight line in a coordinate plane. The foundation of line equations often focuses on the slope and position they convey. In this context, creating a line equation requires understanding the slope-intercept form, which can be expressed as:

• y = mx + b - where y is the dependent variable calculating for any x.
• m is the slope indicating the steepness of the line.
• b is the y-intercept, which tells where the line crosses the y-axis.

Writing an equation involves substituting the known slope, and once calculated, the y-intercept. This concise formula expedites graphing lines by specifying their precise behavior across a graph. Through it, you see how the slope, alongside the intercept, develops a linear path.

y-intercept

The y-intercept of a line is the point where the line crosses the y-axis in a graph. This point is crucial because it reveals a specific location on the vertical axis that the line passes through. When we have the line's equation in slope-intercept form as:

• y = mx + b

The coefficient b represents the y-intercept.Knowing the y-intercept allows us to quickly plot the line on a graph. It's especially useful when combined with the slope m. By plotting the y-intercept and using the slope, we can accurately draw the entire line.For example, in the line equation \[ y = \frac{1}{2}x + 5 \]the y-intercept is 5, meaning the line crosses the y-axis at the point (0, 5). Thus, a y-intercept simplifies the line's initial plotting on the graph.

point-slope formula

The point-slope formula is a crucial method to find the equation of a line when given a point on the line and its slope. This formula is especially useful to derive equations directly without knowing the y-intercept in advance. The general form of the point-slope formula is:

• y - y_1 = m(x - x_1) - where the point \((x_1, y_1)\) specifies a point on the line.
• m is the slope of the line.

To apply point-slope, start by plugging in the known slope and the given point's coordinates into the formula. Once simplified, you obtain the y-intercept, which can be used to achieve the slope-intercept form. For instance, using \((x_1, y_1) = (-6, 2)\) with a slope \[ m=\frac{1}{2}\], the formula yields:\[ y - 2 = \frac{1}{2}(x + 6) \] Simplifying this leads to finding the y-intercept and eventually transforming it into the slope-intercept form. The point-slope formula is a versatile tool that simplifies the process of formulating a line equation from basic geometric information.