Problem 52 - Algebra and Trigonometry

Question

Find the slope and \(y\)-intercept of the line and draw its graph. \(4 x+5 y=10\)

Step-by-Step Solution

Answer

Slope: \(-\frac{4}{5}\), y-intercept: 2; Graph passes through (0, 2) and (5, -2).

Step 1: Rewrite the Equation in 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. We start with the equation \( 4x + 5y = 10 \). We need to solve for \( y \). First, subtract \( 4x \) from both sides to isolate the \( y \)-term: \( 5y = -4x + 10 \).

Step 2: Solve for y

Now, divide each term by 5 to solve for \( y \): \( y = \frac{-4}{5}x + 2 \). This equation is now in the slope-intercept form \( y = mx + b \).

Step 3: Identify the Slope and y-intercept

From the equation \( y = \frac{-4}{5}x + 2 \), we can identify that the slope \( m \) is \( \frac{-4}{5} \) and the \( y \)-intercept \( b \) is \( 2 \).

Step 4: Plot the Graph

To draw the graph, start by plotting the \( y \)-intercept \( (0, 2) \) on the y-axis. Then use the slope \( \frac{-4}{5} \) to find the next point: from \( (0, 2) \), move down 4 units and right 5 units to point \( (5, -2) \). Draw a line that passes through these two points.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a specific way to write the equation of a line. It is commonly used in algebra because it clearly shows the slope and the y-intercept of a line. This form is expressed as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) denotes the y-intercept. By using this form, you can easily identify key characteristics of a line and graph it accordingly.

• Slope (\( m \)): The coefficient of \( x \), which tells us how steep the line is and in which direction it moves.
• Y-intercept (\( b \)): The constant term, indicating where the line crosses the y-axis.

Transforming a linear equation, such as \( 4x + 5y = 10 \), into slope-intercept form involves isolating \( y \). This is done by moving terms involving \( x \) to the opposite side of the equation and adjusting coefficients to solve for \( y \). The outcome will neatly present both the slope and y-intercept, making graphing the line straightforward.

Slope

Slope is a fundamental concept that describes how a line slants or inclines. Mathematically, it is the ratio of the change in the \( y \)-coordinates to the change in the \( x \)-coordinates between any two points on the line. This ratio is often referred to as "rise over run."
An equation's slope offers insight into:

• Direction: A positive slope means the line inclines upwards as it moves from left to right, while a negative slope indicates a downward inclination.
• Steepness: The greater the absolute value of the slope, the steeper the line.

For example, in the equation \( y = \frac{-4}{5}x + 2 \), the slope is \( \frac{-4}{5} \). This tells us that for every 5 units the line moves horizontally to the right, it drops by 4 units vertically. Understanding and calculating the slope is crucial for graphing and interpreting linear equations.

Y-intercept

The y-intercept of a line is the point where it crosses the y-axis. It is a valuable piece of information when graphing a line or understanding its behavior. In the slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept.
When you set \( x = 0 \) in a linear equation, you find the y-intercept. It is essentially the value of \( y \) when the line intersects the y-axis, providing a starting point for graphing.
In the equation \( y = \frac{-4}{5}x + 2 \), the y-intercept is \( 2 \). This means that when \( x = 0 \), \( y = 2 \). Therefore, the line crosses the y-axis at the coordinate \((0, 2)\). Understanding the y-intercept helps in sketching the initial point on the graph and serves as a reference when using the slope to construct the rest of the line.