Problem 28 - Elementary Algebra

Question

Find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. (Objective 1a) \(m=-\frac{5}{7}\) and \(b=-1\)

Step-by-Step Solution

Answer

The equation is \( y = -\frac{5}{7}x - 1 \).

Step 1: Understand Slope-Intercept Form

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

Step 2: Substitute Given Values

Substitute the given values into the slope-intercept form. Here, the slope \( m = -\frac{5}{7} \) and the y-intercept \( b = -1 \). The equation becomes \( y = -\frac{5}{7}x - 1 \).

Step 3: Verify the Equation

Check to ensure that both the slope \( m \) and the y-intercept \( b \) have been correctly substituted into the equation, which should be \( y = -\frac{5}{7}x - 1 \).

Key Concepts

Understanding Linear Equations

Linear equations are fundamental in algebra and are used to represent straight lines on a graph. In a linear equation, each term is either a constant or the product of a constant and a single variable. The general form of a linear equation in two variables, \( x \) and \( y \), is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. However, in slope-intercept form, which is \( y = mx + b \), the equation is solved for \( y \) in terms of \( x \). This form is practical for quickly identifying the slope and the y-intercept of the line, which helps in graphing the equation.

• The slope-intercept form gives a clear picture of how the line behaves on a graph.
• Linear equations can model real-life relationships where changes occur at a constant rate.

The Slope of a Line

The slope of a line is a number that describes both the direction and the steepness of the line. The slope is denoted by \( m \) in the slope-intercept equation \( y = mx + b \). The slope defines how much \( y \) increases (or decreases) as \( x \) is increased by a unit.

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• The formula for calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

In the context of our example, the given slope is \(-\frac{5}{7}\). This indicates that for every 7 units you move to the right along the \( x \)-axis, the \( y \)-value decreases by 5 units.

Y-Intercept of a Line

The y-intercept is the point where the line crosses the y-axis. It is represented by \( b \) in the equation \( y = mx + b \). The y-intercept is significant because it tells you the value of \( y \) when \( x \) is zero.

• The y-intercept is an essential feature used to plot the line, allowing you to draw the graph with the y-intercept as a starting point.
• In our example equation, the y-intercept is \(-1\), meaning the line crosses the y-axis at the point \( (0, -1) \).

By knowing the y-intercept, you can place the line in the correct position on a graph, affirming how the slope further defines its direction.