Problem 75 - Algebra A Combined Function

Question

Write an equation in standard form of the line that contains the point (-1,2) and is parallel to (has the same slope as) the line \(y=3 x-1\)

Step-by-Step Solution

Answer

Equation in standard form is: 3x - y = -5.

Step 1: Understand the problem

We need to find an equation of a line that passes through the point (-1, 2) and is parallel to another line given by the equation \(y = 3x - 1\). The known line has a slope of 3.

Step 2: Determine the slope

The slope of the line we are looking for must be the same as the line \(y = 3x - 1\). Therefore, the slope \(m\) is 3.

Step 3: Use the point-slope form equation

The point-slope form equation of a line is \(y - y_1 = m(x - x_1)\). Given point \((-1, 2)\) and slope \(m = 3\), substitute into the formula: \(y - 2 = 3(x + 1)\).

Step 4: Simplify the equation

Expand the equation: \(y - 2 = 3x + 3\). Then, isolate \(y\) by adding 2 to both sides: \(y = 3x + 5\).

Step 5: Convert to standard form

Standard form of a line is \(Ax + By = C\). Rearrange \(y = 3x + 5\) to \(-3x + y = 5\) by subtracting \(3x\) from both sides. For standard form, \(A\) should be positive, so multiply the entire equation by -1 to get \(3x - y = -5\).

Key Concepts

Point-Slope Form

The point-slope form is an important concept when dealing with linear equations, specifically for lines. It is called the "point-slope" form because it gives you a way to write the equation of a line when you know:

• One point on the line (let's call it \((x_1, y_1)\))
• The slope \(m\) of the line

The formula for the point-slope form is:\[y - y_1 = m(x - x_1)\]This formula helps you write the equation of a line quickly and easily. When using it, replace \(x_1\) and \(y_1\) with the coordinates of the point and \(m\) with the slope you are using.
This form is especially useful when given a point and the slope, making it convenient for problems like the one in the exercise.

Parallel Lines

Parallel lines are lines in the same plane that never intersect. They always maintain the same distance apart, no matter how far they are extended in either direction.
A key property of parallel lines is that they have the same slope.
This means if you're looking to determine if two lines are parallel, check their slopes. If the slopes are equal, the lines are parallel.In the context of the exercise, since we have a line \(y = 3x - 1\), it has a slope of 3. Any line that is parallel to this line must also have a slope of 3.
This property allowed us to use the slope directly in the point-slope form equation to find the desired parallel line.

Slope-Intercept Form

The slope-intercept form is a popular and straightforward way to write equations of a line. The form is particularly useful when you want to quickly understand the characteristics of a line, such as its slope and where it crosses the y-axis.
The formula for the slope-intercept form is:\[y = mx + b\]In this equation:

• \(m\) is the slope of the line
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis

For the line in the exercise, we transformed the given point-slope equation into the slope-intercept form, resulting in \(y = 3x + 5\).
This tells us that the line has a slope of 3 and crosses the y-axis at (0, 5). Understanding and converting between forms is vital in analyzing linear equations and finding their standard form representation.