Problem 60 - Algebra 1

Question

Write an equation of the line in slope-intercept form that passes through the two points, or passes through the point and has the given slope. $$(5,1), m=5$$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = 5x - 24\).

Step 1: Understanding the formula

The slope-intercept form of a line is given by the equation \(y = mx + b\). In this equation, \(m\) represents the slope of the line, and \(b\) represents the y-intercept, which is the point at which the line intersects the y-axis.

Step 2: Substituting the Given Values

Given that the slope of the line \(m = 5\) and one point on the line is \((5,1)\), we can substitute these values into the standard slope equation. By doing this, we now have the equation \(1 = 5 * 5 + b.\)

Step 3: Solving for the y-intercept

Solving for \(b\) in the equation from Step 2 gives us \( b = 1 - 5 * 5\). Thus, \(b = -24\).

Step 4: Writing the Final Equation

Substituting the values of \(m\) and \(b\) into the equation \(y = mx + b\), we get the final equation of the line: \(y = 5x - 24\).\n

Key Concepts

Equation of a Line

In mathematics, an equation of a line is a way to represent a straight line using numbers and symbols. One of the most common forms of this equation is the slope-intercept form, which is expressed as \(y = mx + b\). This particular equation tells us about the line's slope, which is the steepness, and the y-intercept, the point where it crosses the y-axis.

There are a few different ways to write equations for lines, but slope-intercept form is widely used because it clearly displays the line's slope and y-intercept, making it easy to understand and use.

In this form:

• The variable \(m\) represents the slope, indicating how steep or flat the line is.
• The variable \(b\) represents the y-intercept, showing where the line crosses the y-axis.

Knowing the equation of a line allows us to create a graph of it or use it in various geometric problems, such as finding where two lines intersect.

Slope

The slope, represented by the letter \(m\) in the line equation, describes how steep a line is. You can think of the slope as the line's angle compared to the horizontal axis, the x-axis. It tells us how much the line rises or falls as it moves from one point to another.

To calculate the slope between two points, you can use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

This formula takes two points with coordinates \((x_1, y_1)\) and \((x_2, y_2)\), and finds the difference in the y-values and divides it by the difference in the x-values. This gives the rate at which the line rises (or falls) per each unit of horizontal move.

Here's what to remember about slope:

• Positive slope: the line rises from left to right.
• Negative slope: the line falls from left to right.
• Zero slope: the line is horizontal and has no vertical change.
• Undefined slope: the line is vertical and there is no horizontal change.

These are all essential in understanding how the slope affects the shape and direction of a line.

Y-Intercept

The y-intercept, symbolized by \(b\) in the slope-intercept form \(y = mx + b\), is the point where the line crosses the y-axis. It tells us exactly where the line hits the vertical y-axis when the x-value is zero. This point is significant because it marks a familiar landmark on the coordinate grid.

To find the y-intercept, simply set \(x=0\) in the equation of the line and solve for \(y\). In the slope-intercept form, \(b\) has already been isolated, so identifying it in a given equation is straightforward.

Consider the characteristics of y-intercept:

• It provides an easy starting point for graphing the equation.
• It helps to understand the line’s relationship with the y-axis.
• In real-world problems, it can represent a starting value before changes occur, like initial conditions in growth or decay scenarios.

Understanding the y-intercept helps us visualize and analyze a line's position and orientation on a graph effectively.