Problem 24 - Algebra A Combined Function

Question

Find an equation of each line with the given slope that passes through the given point. Write the equation in the form $A x+B y=C. $$ m=4 ; \quad(1,3) $$

Step-by-Step Solution

Answer

The line equation in standard form is \(4x - y = 1\).

Step 1: Understand the Point-Slope Form

Since we know the slope \(m\) and a point \((x_1, y_1)\) that the line passes through, we can use the point-slope form of a line: \[ y - y_1 = m(x - x_1) \] In this case, the slope \(m\) is given as 4, and the point \((1, 3)\) is given. This means \(x_1 = 1\) and \(y_1 = 3\).

Step 2: Substitute the Values

Substitute the given values into the point-slope form: \[ y - 3 = 4(x - 1) \]

Step 3: Simplify the Equation

Expand the equation from the previous step: \[ y - 3 = 4x - 4 \] To simplify, add 3 to both sides to isolate \(y\): \[ y = 4x - 1 \]

Step 4: Convert to Standard Form

Standard form of a line is \(Ax + By = C\). To convert the equation \(y = 4x - 1\) into standard form, rearrange it: \[ 4x - y = 1 \] In this form, \(A = 4\), \(B = -1\), and \(C = 1\).

Key Concepts

Point-Slope Form

The point-slope form of a line is a valuable tool when you know the slope and a specific point the line passes through. This form is written as \( y - y_1 = m(x - x_1) \). Here, \( m \) represents the slope of the line, and \((x_1, y_1)\) is the known point.

To use the point-slope form, you'll substitute the slope and the point's coordinates into this equation. For instance, if you are given a slope \( m = 4 \) and a point \((1, 3)\), you would replace \( m \) with 4, \( x_1 \) with 1, and \( y_1 \) with 3 in the formula:

• Start with: \( y - y_1 = m(x - x_1) \)
• Substitute to get: \( y - 3 = 4(x - 1) \)

This form is excellent for quickly creating an equation of a line without needing to first solve for a variable.

Standard Form of a Line

The standard form of a line is expressed as \( Ax + By = C \), where \(A\), \(B\), and \(C\) are integers. This format is often used because it's straightforward for solving systems of equations.

To convert an equation from another form into the standard form, you need to manipulate it so it fits this pattern. From the exercise, you had the equation \( y = 4x - 1 \) in slope-intercept form. To transform it into standard form:

• Rearrange the terms: \( 4x - y = 1 \)

In this equation, \( A = 4 \), \( B = -1 \), and \( C = 1 \). The standard form provides a clear way to identify the relationship between \( x \) and \( y \). It's also useful in finding intercepts and graphing the line.

Slope-Intercept Form

The slope-intercept form of a line is one of the simplest and most intuitive ways to write a linear equation. It is presented as \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept, meaning the point where the line crosses the y-axis.

In the exercise, by simplifying the point-slope form, you arrived at \( y = 4x - 1 \). Here:

• The slope \( m \) is 4.
• The y-intercept \( b \) is -1.

This format allows you to easily see how the line moves across the graph:

• The slope of 4 means that for every unit increase in \( x \), \( y \) increases by 4 units.
• The y-intercept of -1 tells you that the line crosses the y-axis at the point \((0, -1)\).

This form is incredibly helpful for quickly sketching a line on a graph based on its slope and y-intercept.