Problem 58 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line that passes through the point \((2,-1)\) and is parallel to the line whose equation is \(4 x-3 y=6\)

Step-by-Step Solution

Answer

The equation of the line is \( y = \frac{4}{3}x - \frac{11}{3} \).

Step 1 - Identify the slope of the given line

The equation of the given line is \( 4x - 3y = 6 \). First, rewrite this equation in the slope-intercept form \( y = mx + b \) to find its slope. Solve for \( y \) by isolating it on one side of the equation:\[ 4x - 3y = 6 \]\[ -3y = -4x + 6 \]\[ y = \frac{4}{3}x - 2 \]So, the slope \( m \) of the given line is \( \frac{4}{3} \).

Step 2 - Determine the slope of the parallel line

Since parallel lines have the same slope, the line passing through the point \((2, -1)\) will also have a slope of \( \frac{4}{3} \).

Step 3 - Use the point-slope form

The point-slope form of the equation of a line is given by:\[ y - y_1 = m(x - x_1) \]Here, \( (x_1, y_1) = (2, -1) \) and \( m = \frac{4}{3} \). Substitute these values into the equation:\[ y - (-1) = \frac{4}{3}(x - 2) \]\[ y + 1 = \frac{4}{3}(x - 2) \]

Step 4 - Simplify the equation

Distribute \( \frac{4}{3} \) on the right-hand side:\[ y + 1 = \frac{4}{3}x - \frac{8}{3} \]Subtract 1 from both sides:\[ y = \frac{4}{3}x - \frac{8}{3} - 1 \]\[ y = \frac{4}{3}x - \frac{8}{3} - \frac{3}{3} \]\[ y = \frac{4}{3}x - \frac{11}{3} \]So, the equation of the line is:\[ y = \frac{4}{3}x - \frac{11}{3} \]

Key Concepts

slope-intercept form

The slope-intercept form of a line's equation is very handy. It's expressed as \( y = mx + b \), where

1. \( m \): the slope of the line, showing how steep it is.
2. \( b \): the y-intercept, marking where the line crosses the y-axis.

For instance, in our exercise, we transformed \( 4x - 3y = 6 \) into slope-intercept form to find the slope. Here's how to do it:

1. Isolate \( y \) on one side of the equation: \( -3y = -4x + 6 \).
2. Divide by \( -3 \): \( y = \frac{4}{3}x - 2 \).

Therefore, the slope \( m \) is \( \frac{4}{3} \). This is key because parallel lines share the same slope. Transitioning to slope-intercept form gives us a direct view of a line's slope and intercept.

point-slope form

The point-slope form is another way to write the equation of a line. It's particularly useful when you know a point on the line and its slope. The formula is \( y - y_1 = m(x - x_1) \), where

1. \( (x_1, y_1) \): a point on the line, using its coordinates.
2. \( m \): the slope of the line.

In our example, we need the line passing through \( (2, -1) \) with a slope of \( \frac{4}{3} \).

1. Start with the point-slope form: \( y - (-1) = \frac{4}{3}(x - 2) \).
2. Simplify: \( y + 1 = \frac{4}{3}(x - 2) \).
3. Distribute \( \frac{4}{3} \): \( y + 1 = \frac{4}{3}x - \frac{8}{3} \).
4. Subtract \( 1 \): \( y = \frac{4}{3}x - \frac{11}{3} \).

The point-slope form bridges the gap when we know a specific point and the slope, enabling an easy conversion into standard forms of equations.

parallel lines

Understanding parallel lines is crucial. These lines never cross and have identical slopes. The rule is simple: if two lines are parallel, their slopes \( m \) must be equal. Using our exercise:

1. Given line's slope in slope-intercept form \( y = \frac{4}{3}x - 2 \) is \( \frac{4}{3} \).
2. The parallel line must also possess a slope of \( \frac{4}{3} \).

Hence, for parallelism, equate the slopes. Moving forward: Identifying slopes ensures we write equations correctly for parallel conditions. This same-slope attribute eliminates guesswork, consolidating our understanding of line relationships. By mastering the characteristic of parallel lines, solving related problems becomes straightforward and less daunting.