Problem 55 - Algebra 1

Question

Write an equation of the line that passes through the point and has the given slope. Use slope-intercept form. $$ (1,-5), m=4 $$

Step-by-Step Solution

Answer

The equation of the line is \(y = 4x - 9\).

Step 1: Identify the given values

Unlock the problem by identifying the given values. Here, we have a point (1, -5) and the slope, m = 4.

Step 2: Apply the point-slope formula

Using the point-slope formula for a line, we have \(y - y1 = m(x - x1)\). Substituting the slope m = 4 and the point (1, -5), we obtain: \(y - (-5) = 4(x - 1)\). This simplifies further to \(y + 5 = 4x - 4\).

Step 3: Transform to slope-intercept form

This equation can now be transformed into slope-intercept form \(y = mx + b\) by subtracting 5 from both sides. The final result is: \(y = 4x - 9\).

Key Concepts

Equation of a Line

Understanding the equation of a line is crucial for describing linear relationships. The slope-intercept form is one of the most commonly used representations of a line in algebra. It's given by the formula \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept, the point where the line crosses the y-axis. This form is particularly useful because it's straightforward to graph and to interpret: the slope tells us how steep the line is, while the y-intercept provides a starting point on the graph.

For a line passing through a specific point with a given slope, as in the exercise above, we can plug these values into the slope-intercept form to create the specific equation for that line. Hence, for the point (1, -5) with a slope \( m = 4 \), we eventually derive the equation \( y = 4x - 9 \).

Point-Slope Formula

The point-slope formula is a powerful tool for writing the equation of a line when you know a point the line passes through and the slope. The formula is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point on the line, and \( m \) is the slope. This formula is derived from the definition of the slope and is a direct embodiment of the concept: change in y divided by change in x.

To use this formula, substitute the known point and slope into the equation. For example, the point (1, -5) and slope \( m = 4 \) gives us \( y + 5 = 4(x - 1) \). Simplifying this expression leads to an equation of the line in slope-intercept form.

Linear Equations

Linear equations create the foundation for studying relationships between two quantities. These equations can always be manipulated into the slope-intercept form, providing a clear visual representation on a graph as straight lines. The general form \( ax + by = c \) highlights the principle that any linear equation is solvable for \( y \) in terms of \( x \).

When graphing, every point on the line satisfies the equation, which means that for every x-value, there is a corresponding y-value that makes the equation true. The simplicity of linear equations makes them a primary focus in algebra education, forming the basis for more complex mathematical concepts.

Algebraic Slope

The concept of slope in algebra is a measure of the steepness of a line. It is a crucial concept since it conveys the rate of change between the x and y values. Mathematically, the slope \( m \) is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

A positive slope means the line slants upwards from left to right, while a negative slope slants downwards. A slope of zero indicates a horizontal line, and an undefined slope represents a vertical line. In the exercise, the slope is 4, indicating for every unit the x-value increases, the y-value increases by four units, forming a relatively steep line.