Problem 90 - Algebra 2

Question

A line passes through points \(K(4,4)\) and \(W(-2,10) .\) a. Write an equation for the line in the form \(A x+B y=C\) . Show your work. b. Find the \(x-\) and \(y\) -intercepts.

Step-by-Step Solution

Answer

The equation of the line in the form Ax + By = C is 1*x + 1*y = 8. The x-intercept is at (8, 0), and the y-intercept is at (0, 8).

Step 1: Find the Slope of the Line

Calculate the slope (m) of the line using the two given points K(4,4) and W(-2,10). The slope is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substitute the coordinates of the points into the formula to get:\[ m = \frac{10 - 4}{-2 - 4} \]\[ m = \frac{6}{-6} \]\[ m = -1 \]

Step 2: Write the Slope-Intercept Form of the Equation

Using the slope and one of the given points, write the equation of the line in slope-intercept form (y = m*x + b). Using point K(4,4) and the slope -1:\[ y - y_1 = m(x - x_1) \]\[ y - 4 = -1(x - 4) \]Distribute the slope and simplify:\[ y - 4 = -x + 4 \]\[ y = -x + 8 \]

Step 3: Convert to General Form

Convert the slope-intercept form to the general form, A*x + B*y = C. Start by adding x to both sides and subtracting 8 from both sides to collect like terms:\[ x + y - 8 = 0 \]The general form is:\[ x + y = 8 \]This is equivalent to:\[ 1*x + 1*y = 8 \]which fits the form A*x + B*y = C.

Step 4: Find the x-Intercept

Set y to 0 in the general form equation to find the x-intercept:\[ 1*x + 1*0 = 8 \]\[ x = 8 \]So the x-intercept is at the point (8, 0).

Step 5: Find the y-Intercept

Set x to 0 in the general form equation to find the y-intercept:\[ 1*0 + 1*y = 8 \]\[ y = 8 \]So the y-intercept is at the point (0, 8).

Key Concepts

Slope Formula

Understanding the slope of a line is key in algebra and helps describe the direction and steepness of the line. The slope formula is a mathematical way of finding this slope when given two points on the line. The formula is expressed as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Where \( m \) is the slope, and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points on the line. The slope is calculated by subtracting the y-coordinate of the first point from the y-coordinate of the second point and dividing that by the difference in the x-coordinates of both points. If the slope is positive, the line rises from left to right; if negative, it falls; a zero slope means the line is horizontal, and an undefined slope (division by zero) indicates a vertical line.

For instance, consider two points, K(4,4) and W(-2,10). The slope calculated using these points is \( -1 \), showing that for every unit we move right, we move 1 unit down, indicating a falling line from left to right.

Slope-Intercept Form

The slope-intercept form is one of the simplest ways to express the equation of a straight line. It is given by the equation:
\[ y = mx + b \]
This equation tells us that when you have a slope (\( m \)) and the y-intercept (\( b \)), you can easily graph the entire line. The slope, as previously discussed, shows how steep the line is, while the y-intercept shows where the line crosses the y-axis. When given a slope and any point on the line, you can find \( b \), by rearranging and solving the equation for \( y \) using the point's coordinates. For the two points, K(4,4) and W(-2,10), the equation in slope-intercept form for this line is \( y = -x + 8 \), indicating a slope of -1 and a y-intercept at (0,8).

General Form of a Line

The general form of a line's equation is the most standardized linear equation form and is expressed as:
\[ Ax + By = C \]
Here \( A \), \( B \), and \( C \) are integers, and the equation makes no explicit indication of the slope or y-intercept. This form is typically used in higher mathematics and computer algorithms as it is more suitable for various calculations and systems of equations.

The slope-intercept form \( y = -x + 8 \) can be rewritten into the general form by rearranging the terms to standardize them as \( x + y = 8 \). From this form, you can still derive other characteristics of the line, like its slope and intercepts when necessary.

X-Intercept

The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. To find the x-intercept, you can set the y-coordinate to zero in the equation of a line and solve for x. This is an important characteristic of a line as it provides a point of reference for graphing.

Using the general form \( x + y = 8 \), setting \( y \) to zero gives us \( x = 8 \), which means the line crosses the x-axis at the point (8, 0). Notably, a horizontal line has no x-intercept unless it lies on the x-axis itself.

Y-Intercept

Similarly, the y-intercept is where the line meets the y-axis, and at this coordinate, the x-value is zero. It is found by setting the x-coordinate to zero and solving for y in the line's equation. This characteristic is crucial as it helps identify the starting point of the line on a graph.

Continuing with our line equation in general form \( x + y = 8 \), when we set \( x \) to zero, it simplifies to \( y = 8 \). Therefore, the line crosses the y-axis at the point (0, 8). The y-intercept is particularly emphasized in the slope-intercept form of a line since it's directly given as part of the equation.