Problem 43 - Algebra 1

Question

Write an equation of the line that passes through the points. (2,5),(6,4)

Step-by-Step Solution

Answer

The equation of the line that passes through the points (2,5) and (6,4) is y = -1/4x + 6

Step 1: Calculate the slope

Use two points from the line to determine the slope. These points are (2,5) and (6,4). Calculate the slope (m) using the formula m = (y2-y1) / (x2-x1). Substitute the corresponding values into the formula to get m = (4 - 5) / (6 - 2) = -1/4.

Step 2: Calculate the y-intercept

Substitute the calculated slope and any point, for instance (2,5), into the formula y = mx + b to solve for the y-intercept (b). Hence, the equation becomes 5 = -1/4 * 2 + b, which simplifies to b = 6 after solving.

Step 3: Write the equation of the line

Substitute the slope and y-intercept values into the line equation y = mx + b to get y = -1/4x + 6

Key Concepts

Slope

The slope is a fundamental concept in linear equations that measures how steep a line is. Think of it as the line's 'tilt'. It tells us how much the y-value (vertical) of a point on the line will change with a change in the x-value (horizontal). To find the slope, we use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\)\; are two different points on the line.

• If the slope is positive, the line ascends from left to right.
• If the slope is negative, the line descends from left to right.
• A slope of zero indicates a horizontal line, while an undefined slope means the line is vertical.

In the given exercise, by plugging in the points \((2, 5)\) and \((6, 4)\), the slope \( m \) is calculated as \(-\frac{1}{4}\). This negative value means that our line is descending as it moves from left to right.

Y-intercept

The y-intercept of a line refers to the point where the line crosses the y-axis. This is a critical component of the line equation \( y = mx + b \). The y-intercept is represented by \( b \) in this equation.

• If \( b = 6 \), for instance, the line touches the y-axis at \( y = 6 \).
• The y-intercept provides a starting point for the line on a graph, as \( x \) is always zero at this intercept point.

In our exercise, once the slope was calculated, we used the point \((2, 5)\) and the slope \(-1/4\) to find \( b \). Solving the equation \(5 = -\frac{1}{4}(2) + b\), we determine that \( b = 6 \). This calculation demonstrates where the line intersects the y-axis, providing an anchor for drawing the line.

Point-Slope Form

The point-slope form is another way to express linear equations. It is especially useful when you have a known slope and a single point through which the line passes. It is written as \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the point on the line and \( m \) is the slope.

• This form highlights the slope directly and pinpoints a specific line derivation.
• From point-slope form, you can easily transform the equation to slope-intercept form \( y = mx + b \).

In the exercise, although we transitioned directly to the slope-intercept form, initially recognizing our first equation as point-slope could streamline understanding. Using the point \((2, 5)\) makes the equation \( y - 5 = -\frac{1}{4}(x - 2) \), which can translate back into the familiar slope-intercept form \( y = -\frac{1}{4}x + 6 \). Understanding both formats can offer flexibility in solving different types of linear equation problems.