Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(29-34\), find an equation for each line. Then write your answer in the form \(A x+B y+C=0\). With \(y\)-intercept 3 and slope 2

Step-by-Step Solution

Answer

The standard form is \(2x - y = -3\).

Step 1: Identify the Given Information

We are asked to find an equation of a line with a given slope and y-intercept. The slope (m) is 2 and the y-intercept (c) is 3.

Step 2: Formulate the Slope-Intercept Equation

Recall the formula for the slope-intercept form of a line: \[ y = mx + c \]Substitute the given slope and y-intercept: \[ y = 2x + 3 \]

Step 3: Convert to Standard Form

We need to convert the slope-intercept form to the standard form \(Ax + By + C = 0\). Start by rearranging:\[ y = 2x + 3 \]Subtract 2x from both sides:\[ -2x + y = 3 \]

Step 4: Finalize the Standard Form

The equation \(-2x + y = 3\) isn't quite in standard form since A should be positive. Multiply the whole equation by -1:\[ 2x - y = -3 \]

Key Concepts

slope-intercept form

The slope-intercept form is one of the most commonly used ways to write the equation of a line in algebra. The general format of this equation is \( y = mx + c \). Here, \( m \) represents the slope of the line, and \( c \) stands for the y-intercept. This form is particularly useful for quickly identifying how a line behaves on a graph. To better grasp this concept, let's consider:

• **Slope (\( m \)):** It determines the steepness and the direction of the line. If \( m \) is positive, the line slopes upwards; if negative, it slopes downwards.
• **Y-Intercept (\( c \)):** This is where the line crosses the y-axis, indicating the value of \( y \) when \( x \) is 0.

By using this form, one can quickly sketch the graph of a line just by knowing the slope and the y-intercept.

standard form

The standard form of a linear equation is another essential format, often expressed as \( Ax + By + C = 0 \). In this equation:

• \( A \), \( B \), and \( C \) are integers, with \( A \) typically kept positive.
• Both \( A \) and \( B \) cannot be zero simultaneously because this would not define a line.

The standard form is beneficial because it is versatile and can be used to find intercepts easily:

• To find the x-intercept, set \( y = 0 \) and solve for \( x \).
• Conversely, to find the y-intercept, set \( x = 0 \) and solve for \( y \).

Converting from the slope-intercept form to the standard form often involves rearranging terms and sometimes multiplying through by -1 to ensure \( A \) is positive, making the equation adhere to conventional standards.

y-intercept

The y-intercept is a fundamental part of linear equations, as it indicates the point at which a line crosses the y-axis of a graph. It is denoted by \( c \) in the slope-intercept form \( y = mx + c \). Understanding the y-intercept can simplify graphing and interpreting linear relationships:

• **Visual Interpretation:** If you plot the line on a graph, you would place a point on the y-axis at \( y = c \). This is your starting point when drawing the line.
• **Equation Insight:** The y-intercept helps you see how changes in \( x \) affect \( y \) when the line crosses the y-axis.

The simplicity of identifying a y-intercept makes it appealing for quickly sketching graphs, teaching slope dynamics, and understanding the linearity of two related variables.

slope

The slope of a line is a crucial concept in algebra, contributing significantly to understanding how a line behaves on a graph. Denoted as \( m \) in the equation \( y = mx + c \), the slope indicates the rate at which \( y \) changes as \( x \) changes. It is often referred to as the line's 'steepness'. Here are some key details about slope:

• **Positive Slope:** If \( m > 0 \), the line inclines upward from left to right.
• **Negative Slope:** If \( m < 0 \), the line declines downward from left to right.
• **Zero Slope:** If \( m = 0 \), the line is horizontal, indicating no change in \( y \) regardless of changes in \( x \).
• **Undefined Slope:** Vertical lines have an undefined slope, as the change in \( x \) is zero.Understanding slope helps in predicting how a line extends as \( x \) values increase or decrease, which is pivotal in many applications such as physics and economics.