Problem 69 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. $$ m=-5, \quad(2,-3) $$

Step-by-Step Solution

Answer

Answer: The equation of the line in slope-intercept form is y = -5x + 7.

Step 1: Identify the slope and point

We are given the slope \(m\) and a point \((x_1, y_1)\) on the line: $$ m = -5, \quad (2, -3) $$ The slope-intercept form of a line is: $$ y = mx + b $$

Step 2: Solve for the y-intercept

We'll use the given point \((2, -3)\) and the slope \((-5)\) to solve for the y-intercept, \(b\). Plug the point and the slope into the slope-intercept equation: $$ -3 = -5 (2) + b $$ Now, solve for \(b\): $$ -3 = -10 + b $$ $$ b = 7 $$ So, the y-intercept \(b\) is \(7\).

Step 3: Write the equation in slope-intercept form

Now that we have the slope \(m\) and the y-intercept \(b\), we can write the equation of the line in slope-intercept form: $$ y = -5x + 7 $$ So, the equation of the line in slope-intercept form is: $$ y = -5x + 7 $$

Key Concepts

Equation of a Line

The equation of a line in mathematics is a fundamental concept to understand. One of the most common forms is the slope-intercept form, represented by the formula: \[y = mx + b\]This form allows you to see two important characteristics of a line at a glance: the slope \(m\) and the y-intercept \(b\). The equation of a line helps in graphing by providing a way to precisely plot a line on a coordinate plane. When we have the slope and any point on the line, like in the exercise provided, we can easily determine the complete equation and understand how the line behaves on the graph.

Slope

The slope \((m)\) of a line describes how steep the line is, or its rate of change. It's often referred to as "rise over run," indicating the change in the \(y\)-coordinate divided by the change in the \(x\)-coordinate:

• Positive slope: The line rises as it moves from left to right.
• Negative slope: The line falls as it moves from left to right.
• Zero slope: The line is horizontal.
• Undefined slope: The line is vertical.

In our original exercise, the given slope is \(-5\). This means for every unit increase in \(x\), \(y\) decreases by 5 units. Understanding the slope is crucial for predicting how changes in the independent variable \(x\) affect the dependent variable \(y\).

Y-Intercept

The y-intercept \((b)\) is the point where the line crosses the \(y\)-axis. It is a crucial component of the equation of a line in slope-intercept form as it establishes a starting value for plotting the line on a graph.In the formula \[y = mx + b\],\(b\) represents the y-intercept. When \(x = 0\), the value of \(y\) is equal to \(b\). Using the point and slope provided in the exercise, we calculated the y-intercept as 7. This means the line crosses the \(y\)-axis at the point \((0, 7)\). Recognizing the y-intercept can guide you in accurately sketching the full line on a graph, starting from the point where it directly touches the vertical axis.