Problem 15 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. slope \(-\frac{1}{2},\) passes through \((1,3)\)

Step-by-Step Solution

Answer

The equation is \( y = -\frac{1}{2}x + \frac{7}{2} \).

Step 1: Recall the Slope-Intercept Form

The slope-intercept form of a line is given by the formula \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept of the line.

Step 2: Identify the Given Values

We are given a slope \( m = -\frac{1}{2} \) and a point on the line \( (1, 3) \).

Step 3: Substitute the Slope and Point into the Equation

Substitute \( m = -\frac{1}{2} \), \( x = 1 \), and \( y = 3 \) into the slope-intercept form \( y = mx + b \) to find \( b \).

Step 4: Solve for the Y-Intercept \( b \)

Substitute the values into the equation: \( 3 = -\frac{1}{2}(1) + b \). Simplify to get \( 3 = -\frac{1}{2} + b \). Solve for \( b \) by adding \( \frac{1}{2} \) to both sides: \( 3 + \frac{1}{2} = b \).

Step 5: Simplify the Value of \( b \)

Convert \( 3 + \frac{1}{2} \) to an improper fraction: \( 3 = \frac{6}{2} \), so \( \frac{6}{2} + \frac{1}{2} = \frac{7}{2} \). Thus, \( b = \frac{7}{2} \).

Step 6: Write the Final Equation

With \( m = -\frac{1}{2} \) and \( b = \frac{7}{2} \), the equation of the line in slope-intercept form is \( y = -\frac{1}{2}x + \frac{7}{2} \).

Key Concepts

Linear Equations

Linear equations are the building blocks for understanding lines in mathematics. These equations represent straight lines on the coordinate plane. The basic form of a linear equation is simple: it's a relationship between two variables, usually expressed as \( y = mx + b \). This is known as the slope-intercept form. Here, \( y \) and \( x \) are variables where \( y \) depends on \( x \). The equation tells you what happens to \( y \) when you change \( x \). For example, changing \( x \) by a certain amount will change \( y \) according to the equation's slope \( m \), which we will discuss further.
Linear equations are powerful tools for modeling relationships between quantities. In everyday situations, you'll notice these equations used in economics, physics, and business.
A great way to get started with linear equations is to recognize their components: the slope \( m \), which is the steepness of the line, and the y-intercept \( b \), the point where the line crosses the y-axis.

Y-Intercept

The y-intercept is where the line crosses the y-axis on the coordinate plane. In the equation \( y = mx + b \), \( b \) is the y-intercept. This particular point is essential because it tells you the value of \( y \) when \( x \) is zero.
The y-intercept is easy to spot when you have the equation in slope-intercept form. Simply look at the constant term, and that is your y-intercept! In our example, the equation \( y = -\frac{1}{2}x + \frac{7}{2} \) tells us that the line crosses the y-axis at \( \frac{7}{2} \). This means the graph of this line will meet the y-axis at a little above 3.5 on the \( y \)-axis.
Knowing the y-intercept helps you quickly graph a line and understand where it stands in relation to the axes. It's like knowing the starting point of a race; from there, you can map out where the line goes.

Slope Calculation

Calculating the slope of a line is a crucial part of understanding how the line behaves. The slope, represented by \( m \) in the equation \( y = mx + b \), shows the direction and steepness of the line. It tells us how much \( y \) changes for a given change in \( x \).
The slope is usually calculated as "rise over run," which means the change in \( y \) over the change in \( x \). This can be visualized as the vertical change divided by the horizontal change between two points on the line, expressed as \( m = \frac{\Delta y}{\Delta x} \). In our problem, the slope was already given as \( -\frac{1}{2} \).
When you see a negative slope, like \( -\frac{1}{2} \), it indicates that the line is decreasing; it slopes downward from left to right. A positive slope would mean the opposite, where the line increases as you move from left to right. Understanding the slope helps predict the behavior of the graph as \( x \) changes.