Problem 41 - Elementary Algebra

Question

Find the equation of the line that passes through the points (1,-2) and (0,4) .

Step-by-Step Solution

Answer

Answer: y = -6x + 4

Step 1: Calculate the slope

To calculate the slope (m) of the line, use the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the given points. Plugging in the values, we get the slope: m = (4 - (-2)) / (0 - 1) = 6 / (-1) = -6.

Step 2: Use the point-slope form of the equation

The point-slope form of the equation is: y - y1 = m(x - x1), where (x1, y1) is one of the given points and m is the slope. We can use the point (1, -2) and the slope -6 to write the equation: y - (-2) = -6(x - 1).

Step 3: Simplify the equation

Simplify the equation from step 2: y + 2 = -6x + 6. To write the equation in the slope-intercept form, isolate the variable y: y = -6x + 6 - 2.

Step 4: Write the final equation of the line

The final equation of the line in slope-intercept form is: y = -6x + 4

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is crucial when working with equations of straight lines. The slope essentially tells us how steep the line is. If the slope is positive, the line rises as it moves from left to right. If it’s negative, the line falls.
The formula to find the slope, represented by the letter \( m \), is given by:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line. It's essential to use the formula correctly by maintaining the order of subtraction, so use the second point minus the first point for both the numerator and the denominator.
In the example provided, our slope calculation involves the points \((1, -2)\) and \((0, 4)\):

• Change in \( y \): \( 4 - (-2) = 6 \)
• Change in \( x \): \( 0 - 1 = -1 \)
• Thus, \( m = \frac{6}{-1} = -6 \)

This negative slope indicates the line is moving downwards as it progresses from left to right.

Point-Slope Form

Once you have the slope, you can write the equation of the line using the point-slope form. This form is particularly helpful because it directly incorporates the slope and a specific point on the line.
The point-slope formula is:

• \( y - y_1 = m(x - x_1) \)

This represents a line with slope \( m \) and passing through a particular point \((x_1, y_1)\).
In the given problem, using the point \((1, -2)\) and the slope \( -6 \), the point-slope form becomes:

• \( y - (-2) = -6(x - 1) \)

Simplifying, this equation portrays the line's behavior starting from the known point with the given slope. It's an effective way to establish a line's equation quickly with this data.

Slope-Intercept Form

The slope-intercept form of a line is a popular way to express linear equations because it straightforwardly presents the slope and the y-intercept, making it easy to graph.
The generic form for slope-intercept is:

• \( y = mx + b \)

Where \( m \) represents the slope, and \( b \) is the y-intercept: the point where the line crosses the y-axis (where \( x = 0 \)).
To convert from point-slope to slope-intercept, solve for \( y \). In our example, the conversion from the point-slope equation \( y + 2 = -6x + 6 \) was simplified to:

• \( y = -6x + 4 \)

Here, \( b = 4 \), making \( 4 \) the y-intercept. This means the line crosses the y-axis at \( (0, 4) \). Using this form allows for quick visualization and comparison with other lines, making it a valuable tool in algebra and geometry.