Problem 9 - Algebra A Combined Function

Question

Write an equation of the line passing through the given points. Write the equation in standard form \(A x+B y=C\). See Example 2 . (2,0) and (4,6)

Step-by-Step Solution

Answer

The standard form of the equation is \( 3x - y = 6 \).

Step 1: Find the Slope

To find the slope ( m ) of the line passing through the points (2,0) and (4,6) , use the slope formula: m = \frac{y_2 - y_1}{x_2 - x_1} . Assign: (x_1, y_1) = (2, 0) and (x_2, y_2) = (4, 6) . Substitute these into the formula to get: \[ m = \frac{6 - 0}{4 - 2} = \frac{6}{2} = 3 \] So, the slope is 3 .

Step 2: Use Point-Slope Form to Find the Equation

Now, use the point-slope form equation of the line: y - y_1 = m(x - x_1) . Substitute m = 3 and the point (2, 0) into the formula: \[ y - 0 = 3(x - 2) \] Simplify this to get: \[ y = 3x - 6 \] This is the equation of the line in slope-intercept form.

Step 3: Convert to Standard Form

The goal is to express the equation in the standard form: Ax + By = C . Start from the equation y = 3x - 6 and rearrange it: \[ 3x - y = 6 \] Ensure all coefficients are integers. Since they are, you have the equation in standard form.

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness or inclination. When given two points that lie on a line, we can calculate the slope using the formula:

• Given points ((x_1, y_1)) and ((x_2, y_2)), the slope,((m)), is calculated as:

\[ m = \frac{y_2 - y_1}{x_2 - x_1}\]For example, if we have the points (2,0) and (4,6), we substitute them into the formula:\[m = \frac{6 - 0}{4 - 2} = \frac{6}{2} = 3\]This tells us that the line is rising 3 units vertically for every 1 unit it moves horizontally. This slope value is crucial for writing the equation of the line.

• The steeper the slope, the larger the absolute value of ((m)).
• A positive slope means the line ascends from left to right, while a negative slope indicates it descends.

Understanding slope helps in graphing lines and interpreting linear relationships.

Point-Slope Form

The point-slope form is a useful equation for writing linear equations when you know a point on the line and the slope. The formula is:

• ((y - y_1 = m(x - x_1)), where ((m)) is the slope and ((x_1, y_1)) is a point on the line.

For the example with slope ((m = 3)) and point (2,0), substitute these into the equation:\[y - 0 = 3(x - 2)\]Simplifying gives us the slope-intercept form:\[y = 3x - 6\]This form, ((y = mx + b)), clearly shows the slope and the y-intercept of the line.

• It's especially effective for quick graphing or transforming into other forms.
• Using point-slope form can make calculations quicker, especially when starting from a known point.

It is a flexible form that allows easy manipulation of the line equation.

Standard Form Conversion

After obtaining the line equation through point-slope form, converting to standard form is often desired for further analysis or certain applications. The standard form of a line is:

• ((Ax + By = C) ), where ((A) ), ((B) ), and ((C) ) are integers, and ((A) ) is non-negative.

Starting with the slope-intercept form ((y = 3x - 6) ), we want to eliminate fractions and rearrange:

• First, move ((y) ) to the other side: ((3x - y = 6) ).

Verify that all coefficients (3, -1, and 6) are integers. If necessary, multiply through by an appropriate factor. Here, they already are integers, so no further adjustment is needed.

• The standard form prioritizes organization and is often easier for comparing multiple linear equations.

This robust form allows straightforward algebraic manipulation and is good for solving systems of equations.