Problem 53 - Algebra 1

Question

Find the slope and the \(y\) -intercept of the graph of the equation. Then graph the equation. $$ 4 y+12 x=16 $$

Step-by-Step Solution

Answer

The slope of the equation is -3 and the y-intercept is 4.

Step 1: Rearrange The Equation

To get the equation in slope-intercept form, isolate y. Start by subtracting 12x from both sides which results in: \(4y = -12x + 16\).

Step 2: Extract slope and y-intercept

To get y alone, divide all terms in the equation by 4: \(y = -3x + 4\). Now the equation is in the form \(y = mx + c\). The coefficient of \(x\), which is -3, is the slope (m), and the constant term, which is 4, is the y-intercept (c).

Step 3: Graph The Equation

Start the graph by marking the y-intercept at 4 on the y-axis. From this point, use the slope to find the next point. Since the slope is -3, this means for every 1 unit you move to the right on the x-axis, move down 3 units on the y axis. Then connect the points with a straight line.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra, representing straight lines when plotted on a graph. The most common form of a linear equation is the slope-intercept form, expressed as \( y = mx + c \). Here, \( m \) is the slope and \( c \) is the y-intercept.
Understanding and manipulating these equations is crucial for drawing relationships between two variables. Linear equations can also be presented in different forms, such as the standard form \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.

• The slope \( m \) indicates the steepness and direction of a line.
• The y-intercept \( c \) indicates where the line crosses the y-axis.

To solve linear equations, you often rearrange them into a desired form for easier graphing or analyzing. This can be done through basic algebraic operations such as addition, subtraction, and division.

Graphing Lines

Graphing lines is the process of plotting linear equations on a coordinate plane. It visually represents the relationship between the variables \( x \) and \( y \).
The process starts with identifying the y-intercept and slope from the equation. For example, if you have the equation \( y = -3x + 4 \), the y-intercept is 4, and the slope is -3. Start by plotting the y-intercept on the y-axis.

• Mark the y-intercept point, which tells you where the line will start.
• From the y-intercept, use the slope to determine the line's angle and direction.

With a slope of -3, for every unit moved to the right along the x-axis, move down 3 units along the y-axis. This tells you how the line angles away from the y-intercept. Connect these points to form a straight line.

Slope and Y-intercept

The slope and y-intercept are key features that define the behavior of a linear equation in its slope-intercept form. The slope, represented as \( m \) in equations, shows how much \( y \) changes for a unit change in \( x \).
In the equation \( y = -3x + 4 \), the slope \( -3 \) means that for every increase of 1 in \( x \), \( y \) decreases by 3. This negative sign indicates that the line descends from left to right. Meanwhile, the y-intercept, \( 4 \), tells us that the line crosses the y-axis at the point \((0, 4)\).

• Analyze the slope to understand the line's direction (positive slopes rise, negative slopes fall).
• The y-intercept shows the starting point on the graph.

Grasping these concepts helps in predicting how changes in \( x \) will affect \( y \) and, consequently, the visual representation of the equation on the graph.