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Chapter 2: Problem 48

Write each equation in slope鈥搃ntercept form. Then find the slope and the y-intercept of the line determined by the equation. $$ -2 x-4 y=-12 $$

Short Answer

Expert verified
Slope: \(-\frac{1}{2}\), Y-intercept: \(3\).

Step by step solution

01

Understand the 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. To solve this problem, we need to rearrange the given equation into this form.
02

Isolate the y-variable

Start with the equation: \[ -2x - 4y = -12 \]To isolate \( y \), first add \( 2x \) to both sides:\[ -4y = 2x - 12 \]
03

Solve for y

Divide every term in the equation by \(-4\) to solve for \( y \):\[ y = \frac{2}{-4}x + \frac{-12}{-4} \] Simplifying, we have:\[ y = -\frac{1}{2}x + 3 \]
04

Identify the Slope and Y-intercept

From the equation \( y = -\frac{1}{2}x + 3 \), the slope \( m \) is \(-\frac{1}{2}\) and the y-intercept \( b \) is \( 3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are fundamental in mathematics. They create a straight line when graphed on a coordinate plane. This type of equation can generally be written in the form \( ax + by = c \). It's important because it forms the basis of many algebraic processes and helps us understand relationships between different quantities. The challenge often resides in changing this form into the slope-intercept form, \( y = mx + b \), which is more descriptive of the line's properties:
  • While the original form, like \( -2x - 4y = -12 \), is perfectly valid for linear equations, it doesn't directly tell us the slope and y-intercept.
  • Rewriting the equation to slope-intercept form makes these features more apparent.
  • Whether you are modeling a real-world situation or solving for unknowns in other equations, understanding how to interpret and transform linear equations is vital.
Slope
The slope of a line is a measure of its steepness or tilt, indicating how much one variable changes with respect to another. In the slope-intercept form \( y = mx + b \), \( m \) represents the slope. This is helpful to visualize and understand the relationship between variables.After converting the equation \( -2x - 4y = -12 \) to slope-intercept form, we found that:
  • \( y = -\frac{1}{2}x + 3 \) has a slope of \(-\frac{1}{2}\).
  • This value of \(-\frac{1}{2}\) tells us that for every unit increase in \( x \), \( y \) decreases by half a unit.
  • Slope can be positive, negative, zero or undefined, each providing different interpretations:
    • Positive slopes rise from left to right, negative slopes fall from left to right.
    • A zero slope creates a horizontal line, while an undefined slope indicates a vertical line.
Understanding slope aids in predicting how changes in one variable affect another, which is fundamental in fields like economics, physics, and data science.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. This occurs when \( x = 0 \). In the slope-intercept form, \( y = mx + b \), the value \( b \) signifies the y-intercept, thereby providing an initial value of \( y \) when \( x \) remains at zero.In our example, when the equation was converted to \( y = -\frac{1}{2}x + 3 \):
  • The y-intercept \( b \) is \( 3 \).
  • This means the line crosses the y-axis at the point \((0, 3)\).
  • Understanding the y-intercept provides initial conditions, helping to predict outcomes or set up a context for real-life scenarios.
Recognizing the y-intercept ensures we know where the line begins on a graph, which is key when computing intersections or comparing different linear relationships.

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