Problem 8 - Algebra 1: Concepts and Skills

Question

Write the equation of the line passing through the two points. Show that this line is perpendicular to the given line. $$ (-4,-4),(-2,2) ; y=-\frac{1}{3} x-1 $$

Step-by-Step Solution

Answer

The equation of the line passing through the given points is y = 3x + 8 and it is indeed perpendicular to the line y = -1/3x - 1.

Step 1: Calculate the slope

Given two points, (-4,-4) and (-2,2), the slope (m) of the line passing through these points is calculated using the formula (y2 - y1) / (x2 - x1). Therefore, m = (2 + 4) / (-2 + 4) = 3.

Step 2: Write a formula for the line

Now we use the slope-intercept form y = mx + c to write the equation of the line. We already know m = 3. To find c, we substitute one of the points into the equation. Let's use (-4,-4): -4 = 3*(-4) + c gives c = -4 - (-12) = 8. Thus, the equation of the line is y = 3x + 8.

Step 3: Confirm Perpendicularity

To prove that the line we found is perpendicular to the given line (y = -1/3x - 1). Two lines are perpendicular if and only if the product of their slopes is -1. The slope of the given line is -1/3. The product of the two slopes is therefore 3*(-1/3) = -1, confirming the perpendicularity between the lines.

Key Concepts

Slope of a Line

Understanding the concept of the slope of a line is crucial in algebra and geometry. The slope is a measure of how steep a line is, and it is determined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Specifically, it is calculated using the formula:
\( m = \frac{{y2 - y1}}{{x2 - x1}} \).
In the given exercise, with the points (-4, -4) and (-2,2), the slope calculation would be \( m = \frac{{2 - (-4)}}{{-2 - (-4)}} = \frac{{6}}{{2}} = 3 \).
This means that for every unit you move horizontally, the line rises by 3 units. Hence, a positive slope indicates that the line is moving upward as it goes from left to right, while a negative slope would indicate a downward direction. Zero slope means the line is horizontal, and an undefined slope corresponds to a vertical line.

Slope-Intercept Form

The slope-intercept form is a straightforward way to write a linear equation. This form is given by \( y = mx + c \),
where \( m \) is the slope and \( c \) is the y-intercept, which is the point where the line crosses the y-axis. In other words, \( c \) is the y-value when \( x \) is zero.
After finding the slope of a line between two points, as we did in the previous section, you can then insert one of the points and the slope value back into the slope-intercept form to solve for \( c \). Once you have \( c \), the equation of the line is complete. In our exercise, substituting the point (-4, -4) and the slope 3 into the equation gives us the y-intercept:
\( -4 = 3(-4) + c \), solving this, \( c = 8 \).
The final equation of the line then is \( y = 3x + 8 \). This quick method ensures you can write the linear equation in slope-intercept form with ease and correctly depict the line on a graph.

Perpendicular Lines in Algebra

Perpendicular lines intersect at a right angle (90 degrees) and have an interesting relationship between their slopes. In algebraic terms, two lines are perpendicular if the product of their slopes is -1. Understanding this concept is essential for many geometry problems and proofs.
If a line has a slope of \( m \), the slope of a line perpendicular to it will be \( -\frac{1}{m} \). So considering a line with a slope of -1/3, like the one given in our exercise \( y = -\frac{1}{3}x - 1 \), a line perpendicular to it should have a slope of \( -1 \) divided by \( -\frac{1}{3} \), which equals 3. We previously determined that our line has a slope of 3, meeting this condition. By calculating the product of their slopes \( 3 \times -\frac{1}{3} = -1 \), we confirm that the two lines are indeed perpendicular. With this rule, verifying the perpendicularity between two lines is a straightforward task in algebra.