Problem 53 - Algebra 1

Question

Write an equation of the line that passes through the point and has the given slope. Use slope-intercept form. $$ (3,0), m=-4 $$

Step-by-Step Solution

Answer

The equation of the line is \( y = -4x + 12 \)

Step 1: Identify the given slope and point

The given slope \( m \) is -4, and the provided coordinates of the point are (3,0)

Step 2: Substitute the slope and point into the slope-intercept equation

Substitute \( m = -4 \), \( x = 3 \), and \( y = 0 \) into \( y = mx + b \), giving \( 0 = -4 * 3 + b \)

Step 3: Solve for the y-intercept, b

Rearranging and solving the equation for \( b \) gives \( b = 0 + 12 = 12 \)

Step 4: Write the equation of the line

With \( m = -4 \) and \( b = 12 \), the equation of the line is \( y = -4x + 12 \)

Key Concepts

Slope-Intercept Form

The slope-intercept form is a fundamental expression in algebra for straight-line equations. It is represented as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. The slope \( m \) measures the steepness and direction of the line, indicating how much \( y \) changes for a unit change in \( x \). The y-intercept \( b \) is where the line crosses the y-axis.

Using this form can greatly simplify both the visualization and calculation of a line's behavior on a graph. For instance, in the exercise above, the line passes through the point (3,0) with a slope of -4. By knowing these two pieces of information, we can straightforwardly determine the line's equation in slope-intercept form—which will prove invaluable for graphing or solving various algebraic problems.

Solving for Y-Intercept

When you have a point on the line and the slope, you can solve for the y-intercept \( b \) by rearranging the slope-intercept equation. If we denote the given point's coordinates by \((x_1, y_1)\), the process involves substituting the slope \( m \) and the coordinates into the equation, leading to \( y_1 = mx_1 + b \). After the substitution, you solve for \( b \), the y-intercept, which gives you the exact point where the line cuts through the y-axis.

In our specific exercise, the equation becomes \(0 = -4 \times 3 + b\), which simplifies to \( b = 12 \). Identifying the y-intercept is a critical step in graphing linear equations and forms the starting point for the line on the graph.

Linear Equations

Linear equations are the simplest form of algebraic functions that describe a straight line on a graph. They have one or more variables but must adhere to the first degree, meaning that the variables are not raised to any power other than one. The key properties of a linear equation include a constant rate of change (slope), no curvature (it's straight), and an intersection point with the y-axis (y-intercept).

When graphing, each linear equation will produce a straight line that extends infinitely in both directions, and every point on this line is a solution to the equation. Linear equations are foundational in algebra and are used to model various real-life situations, like predicting trends and understanding relationships between quantities. The exercise we tackled translates into a linear equation that can easily plot a precise line on the Cartesian plane.