Problem 8 - 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 \(-\frac{1}{5} ;\) through (4,-6)

Step-by-Step Solution

Answer

The equation of the line is \( y = -\frac{1}{5}x - \frac{26}{5} \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept, which is where the line crosses the y-axis. We need to find both \( m \) and \( b \) to write the equation.

Step 2: Substitute the Slope

We are given a slope of \( -\frac{1}{5} \). Substitute this into the equation \( y = mx + b \) to get \( y = -\frac{1}{5}x + b \). This will help us in finding the y-intercept \( b \).

Step 3: Use the Point to Solve for b

We need to substitute the given point \((4, -6)\) into the equation \( y = -\frac{1}{5}x + b \). Using \( x = 4 \) and \( y = -6 \), we substitute them in:\[ -6 = -\frac{1}{5}(4) + b \]Simplify the equation:\[ -6 = -\frac{4}{5} + b \]

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

Continuing from the previous step:Add \( \frac{4}{5} \) to both sides of the equation to solve for \( b \):\[ -6 + \frac{4}{5} = b \]Convert \( -6 \) to a fraction: \( -\frac{30}{5} \)\[ -\frac{30}{5} + \frac{4}{5} = b \]\[ b = -\frac{26}{5} \]

Step 5: Write the Final Equation

Now that we have both \( m \) and \( b \), substitute them back into the slope-intercept form \( y = mx + b \):\[ y = -\frac{1}{5}x - \frac{26}{5} \]This is the equation of the line with the given slope that goes through the point \((4, -6)\).

Key Concepts

Equation of a Line

The equation of a line is a powerful tool in algebra that represents a straight line on a coordinate plane. In its simplest form, the equation is expressed as \( y = mx + b \). The two main components of this equation are the slope \( m \) and the y-intercept \( b \). Understanding these components can help in visualizing how the line behaves. The slope \( m \) indicates the steepness and direction of the line, while the y-intercept \( b \) shows where the line crosses the y-axis. By knowing these, we can predict and calculate every point along the line. A line’s equation consolidates an infinite number of points into a concise algebraic expression.

Finding Y-Intercept

Finding the y-intercept \( b \) is crucial when writing the equation of a line in slope-intercept form. The y-intercept is the point where the line crosses the y-axis, meaning it occurs at \( x = 0 \). To find \( b \), you begin with substituting the known values into the equation \( y = mx + b \). If the slope \( m \) and a point \((x, y)\) are known, simply rearrange and solve for \( b \).

• Insert the values of the slope and the coordinates into the equation.
• Simplify the resulting expression by solving for \( b \).

This isolates the y-intercept, allowing you to determine exactly where the line will intersect the y-axis.

Using Given Point for Equations

When given a specific point on the line, such as \((4, -6)\), this information can be used to complete the equation of the line. Inserting the x and y coordinates of the point into the equation \( y = mx + b \) assists in solving for the y-intercept \( b \). The process is straightforward:

• Begin with the equation with the known slope and unknown \( b \), such as \( y = -\frac{1}{5}x + b \).
• Substitute the x-value and y-value from the point into the equation.
• Rearrange the equation to solve for \( b \).

Understanding how to integrate a point into the equation ensures accuracy and helps verify the line passes through this specific point.

Algebraic Problem Solving

Algebraic problem solving involves using algebraic methods to find unknown values. In this context, solving for \( b \) requires manipulating the equation \( y = mx + b \) to isolate \( b \). This involves combining like terms and adjusting both sides of the equation until you have \( b = \) (some value). For example:

• Follow the arithmetic operations to simplify the expressions.
• Convert whole numbers to fractions if needed, to match denominators.
• Ensure all calculations are properly executed, especially when fractions are involved.

These problem-solving skills are widely applicable in mathematics, helping you to solve not only simple line equations but also more complex algebraic expressions.