Problem 22 - Algebra A Combined Function

Question

Mixed Practice Find the slope of each line. See Examples 3 through 6. $$ -5 x+y=10 $$

Step-by-Step Solution

Answer

The slope of the line is 5.

Step 1: Rewrite the equation in slope-intercept form

The slope-intercept form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start with the given equation, \( -5x + y = 10 \). Add \( 5x \) to both sides to solve for \( y \). This gives you \( y = 5x + 10 \).

Step 2: Identify the slope

In the slope-intercept form \( y = mx + b \), the coefficient of \( x \) is the slope. From the equation \( y = 5x + 10 \), identify \( m = 5 \). Thus, the slope of the line is \( 5 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to easily identify the slope and y-intercept of a line. It is written as \( y = mx + b \) where:

• \( y \) is the dependent variable (usually representing vertical position on the graph).
• \( m \) is the slope of the line, showing how steep the line is.
• \( x \) is the independent variable (usually representing horizontal position on the graph).
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

To convert a standard or mixed form equation like \(-5x + y = 10\) into slope-intercept form, you solve for \( y \) in terms of \( x \). In the given exercise, adding \( 5x \) to both sides neatly rearranges the equation into \( y = 5x + 10 \). This transformation centers around isolating \( y \) to one side of the equation, making the slope (\( m \)) and y-intercept (\( b \)) clear.

Linear Equations

Linear equations represent straight lines on a graph and have no variables raised to a power other than one. They are called "linear" because their graph is a straight line. A standard form of a linear equation can include multiple variables and constants, like \(-5x + y = 10\).
To graphically represent these equations, one needs to identify specific properties such as slope and intercepts, which can be quickly detected if the equation is rewritten in slope-intercept form.
This transformation is especially useful because the slope-intercept form directly illustrates the relationship between the variables and how the graph of the equation behaves. Linear equations are foundational in mathematics, used to model real-world relationships where change occurs at a constant rate.

Coefficient of x

In any linear equation of the form \( y = mx + b \), the coefficient \( m \) attached to the \( x \) term indicates the slope of the line. The coefficient of \( x \) is crucial because it quantitatively shows how much \( y \) increases or decreases as \( x \) increases by one unit.

• If \( m \) is positive, the line slopes upwards, indicating a direct relationship.
• If \( m \) is negative, the line slopes downwards, signaling an inverse relationship.
• If \( m \) is zero, the line is horizontal, showing no change regardless of \( x \).

For the example \( y = 5x + 10 \), the coefficient \( 5 \) suggests that for every increase of 1 in \( x \), \( y \) will increase by 5. Understanding the coefficient of \( x \) helps predict and interpret the behavior of the line.