Problem 8 - Algebra 1

Question

Write an equation of the line that passes through the point and has the given slope. Write the equation in slope-intercept form. $$(0,-5), m=0$$

Step-by-Step Solution

Answer

\(y = -5\)

Step 1: Identifying the known quantities

The given point through which the line passes is (0,-5) and the slope of the line \(m=0\). In the slope-intercept form \(y = mx + b\), the slope \(m\) is the 'rise over run' between any two points on the line, and \(b\) is the y-intercept, the point where the line crosses the y-axis.

Step 2: Substituting the slope into the general equation

The given slope \(m=0\). Therefore the equation becomes \(y = 0x + b\).

Step 3: Substituting the point values to find b (y-intercept)

The given point is (0,-5). This falls on the y-axis itself; hence it is also the y-intercept. Substituting x = 0, and y = -5, we get -5 = 0(0) + b, which simplifies to b = -5.

Step 4: Substituting the b value in the equation

By substituting the value of b in the equation we get the final equation \(y = 0x - 5\). However since 0x is 0 for all x, we often simplify this to \(y = -5\).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that define straight lines on a coordinate plane. These equations are of the form \(y = mx + b\), where \(m\) and \(b\) are constants. This standard form makes it easy to identify the characteristics of the line.
Linear equations consist of:

• An independent variable \(x\).
• A dependent variable \(y\).
• A constant \(m\) which represents the slope of the line.
• A constant \(b\) which is called the y-intercept.

Understanding linear equations is crucial because they appear frequently in various real-life applications like calculating speed, predicting future trends, and financial modeling. The beauty of linear equations is their simplicity, as they offer a direct relationship between \(x\) and \(y\), providing a straightforward way to predict values. By mastering linear equations, students can develop a firm foundation for more complex algebraic concepts.

y-intercept

The y-intercept of a linear equation is the point where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by \(b\). It is the value of \(y\) when \(x = 0\). This means when we substitute \(x = 0\) into the equation, the resulting \(y\) coordinate gives us the y-intercept.
For example, in our exercise, the given point is \((0, -5)\). Since the \(x\) value is 0, \(-5\) is the y-intercept. It informs us that the line passes through the y-axis at the point \((0, -5)\), indicating its significance in the equation of the line.
Understanding the y-intercept helps us easily plot graphs and visualize the line. It's an essential entry point, especially when graphing lines, as it gives a starting reference from the y-axis.

Slope

The slope of a line is a measure of its steepness and direction, and is symbolized by \(m\) in the slope-intercept form \(y = mx + b\). The slope is calculated as the "rise" (change in \(y\)) over the "run" (change in \(x\)) between two points on the line.
For a slope of \(0\), like in this exercise, it indicates a completely horizontal line. This means there’s no change in the \(y\) value as \(x\) changes—the line doesn’t rise or fall but stays consistent, and the equation simplifies directly to the form \(y = b\).
Understanding slopes is fundamental:

• Positive slope: the line rises from left to right.
• Negative slope: the line falls from left to right.
• Zero slope: the line is horizontal.
• Undefined slope: the line is vertical.

Grasping the concept of slope can help you determine how a line behaves across a graph, making it crucial in identifying and predicting linear trends.