Problem 32 - Algebra A Combined Function

Question

Find an equation of the line passing through each pair of points. Write the equation in the form $A x+B y=C. $$ (6,2) \text { and }(8,8) $$

Step-by-Step Solution

Answer

The equation in standard form is \( 3x - y = 16 \).

Step 1: Find the Slope

To determine the slope \( m \) of the line, use the formula for slope: \[ m = \frac{y_2-y_1}{x_2-x_1} \]where \((x_1, y_1) = (6, 2)\) and \((x_2, y_2) = (8, 8)\). Substitute these coordinates: \[ m = \frac{8-2}{8-6} = \frac{6}{2} = 3 \] Thus, the slope of the line is \( 3 \).

Step 2: Use Point-Slope Form

With the slope known, utilize the point-slope form of the line's equation: \[ y - y_1 = m(x - x_1) \]Substitute \( m = 3 \), and point \( (6, 2) \) into the equation: \[ y - 2 = 3(x - 6) \] Simplify to get the linear equation: \[ y - 2 = 3x - 18 \] \[ y = 3x - 16 \]

Step 3: Rearrange to Standard Form

The standard form of a linear equation is \( Ax + By = C \). Rearrange the equation from step 2 to this form: Start with: \[ y = 3x - 16 \] Subtract \( 3x \) from both sides: \[ -3x + y = -16 \] Multiply through by \(-1\) to make \( A \) positive: \[ 3x - y = 16 \] The equation in standard form is: \[ 3x - y = 16 \]

Key Concepts

Slope Calculation

Calculating the slope is the first step when determining the equation of a line. The slope, often represented as \( m \), indicates how steep the line is. A key formula to remember for finding the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \( m = \frac{y_2-y_1}{x_2-x_1} \)

Let's break this down:
- The numerator \( y_2-y_1 \) shows the change in the y-coordinates, often called "rise."
- The denominator \( x_2-x_1 \) shows the change in the x-coordinates, also known as "run." By using these changes, you can determine whether the line is rising, falling, horizontal, or vertical. A positive slope means the line rises as it moves from left to right. Alternatively, a negative slope means it falls.
In our example, calculating between points \((6, 2)\) and \((8, 8)\), we get:

• \( m = \frac{8 - 2}{8 - 6} = 3 \)

This indicates a line that rises steeply.

Point-Slope Form

Once you know the slope, you can use the point-slope form to find the equation of a line. The point-slope form is particularly useful because it incorporates a known point on the line and the slope in its structure:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) is a given point on the line, and \( m \) is the slope.In the exercise, we know \( m = 3 \) and the point \( (6, 2) \), so:

• \( y - 2 = 3(x - 6) \)

This equation directly tells how every point \( (x, y) \) on the line relates to the chosen point \((6, 2)\).The advantage of the point-slope form is its simplicity when you have limited information—a single point and the slope can guide you to the full equation.

Standard Form of a Linear Equation

The standard form of a linear equation is written as:

• \( Ax + By = C \)

This presents the equation with integer coefficients, which many find easier to interpret and use, especially in coordinate geometry tasks.After obtaining the slope-intercept form \( y = 3x - 16 \), you can rearrange it to standard form:

• Begin with \( y = 3x - 16 \).
• Subtract \( 3x \) from both sides to get \(-3x + y = -16 \).
• Multiply through by \(-1\):\( 3x - y = 16 \).

Now, the equation \( 3x - y = 16 \) fits the \( Ax + By = C \) format where \( A = 3 \), \( B = -1 \), and \( C = 16 \).This form is especially useful when dealing with systems of linear equations or performing operations that require a standard presentation of the line equation.