Problem 13 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((4,-2)\) with slope 1

Step-by-Step Solution

Answer

\( y = x - 6 \)

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Substitute the Given Slope

We know the slope (\( m \)) is 1. Substitute 1 for \( m \) in the equation: \( y = 1x + b \) or \( y = x + b \).

Step 3: Use the Given Point to Find the Y-Intercept

The line passes through the point \((4, -2)\). Substitute \( x = 4 \) and \( y = -2 \) into the equation \( y = x + b \) to find \( b \).\( -2 = 4 + b \)

Step 4: Solve for the Y-Intercept

Now, solve the equation \( -2 = 4 + b \) for \( b \).Subtract 4 from both sides:\( b = -2 - 4 \)\( b = -6 \)

Step 5: Write the Equation of the Line

Substitute \( b = -6 \) back into the slope-intercept form equation:\( y = x - 6 \).

Key Concepts

slope-intercept form

The slope-intercept form is a simple way to write down the equation of a line. This format makes it easy to identify the slope and the y-intercept right away. The general formula is:
\[ y = mx + b \] Here,

• \textbf{m} is the slope of the line, which tells you how steep the line is.
• \textbf{b} is the y-intercept, which is the point where the line crosses the y-axis.

To use this form, start by substituting the given slope into the equation. This makes it easier to find the y-intercept when you know a point on the line.

slope

The slope (\textbf{m}) is a measure of how steep a line is. It represents the ratio of the rise (the change in y) over the run (the change in x). In mathematical terms:
\[ m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}} \] In our exercise, the slope is given as 1. This means for every 1 unit you move along the x-axis (run), the y-coordinate also changes by 1 unit (rise).
When you have the slope, you can use it to write the initial form of the line equation in the slope-intercept form:
\[ y = mx + b \] By knowing how the slope affects the position and angle of the line, you can better understand and visualize the line's direction.

y-intercept

The y-intercept (\textbf{b}) is where the line crosses the y-axis. At this point, the x-coordinate is always 0. In our exercise, we need to find the y-intercept after substituting the given slope and one point on the line into the equation.
Start with the equation after substituting the slope:
\[ y = x + b \] Next, use the given point (\textbf{4, -2}). Substitute 4 for x and -2 for y into the equation:
\[ -2 = 4 + b \] Solve for \textbf{b} by isolating it on one side of the equation:
\[ b = -2 - 4 \]
\[ b = -6 \] Now that we know \textbf{b} is -6, the full equation of the line is:
\[ y = x - 6 \] This means the line crosses the y-axis at (-6). Knowing the y-intercept helps you plot the line accurately.