Problem 10 - 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 . (3,0) and (7,8)

Step-by-Step Solution

Answer

The equation of the line is \(2x - y = 6\).

Step 1: Find the Slope

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the equation: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Plug in the values from the points (3, 0) and (7, 8):\[m = \frac{8 - 0}{7 - 3} = \frac{8}{4} = 2\].

Step 2: Use Point-Slope Form

With a slope of 2 and one of the points (3,0), use the point-slope form: \(y - y_1 = m(x - x_1)\). Substitute \(m = 2\), \(x_1 = 3\), and \(y_1 = 0\):\[y - 0 = 2(x - 3)\]Simplifying, we get:\[y = 2x - 6\].

Step 3: Convert to Standard Form

Standard form of a linear equation is \(Ax + By = C\). From the equation \(y = 2x - 6\), rearrange terms to obtain:\[2x - y = 6\].

Step 4: Ensure Integer Coefficients

The equation \(2x - y = 6\) already has integer coefficients and is in the form \(Ax + By = C\). Thus, \(A = 2\), \(B = -1\), and \(C = 6\) are integers, confirming the equation is in standard form.

Key Concepts

Slope Calculation

Understanding how to calculate the slope is essential when working with linear equations. The slope represents the steepness or inclination of a line and is computed using two points on the line. This is done with the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line. This formula measures the change in the \(y\)-coordinates relative to the \(x\)-coordinates.
For example, if you have the points (3,0) and (7,8), you substitute these into the formula:

• Calculate the change in \(y\): \(8 - 0 = 8\)
• Calculate the change in \(x\): \(7 - 3 = 4\)
• Divide these changes to find the slope \(m: \frac{8}{4} = 2\)

The positive slope here means the line rises from left to right.

Point-Slope Form

Once you have the slope, the point-slope form is a convenient way to write the equation of a line. This form is expressed as:

• \(y - y_1 = m(x - x_1)\)

Where \(m\) is the slope, and \((x_1, y_1)\) is any specific point on the line. This form is quite flexible because you can use any point on the line, not just the intercepts.
In our example with slope \(m = 2\) and point (3,0), we substitute these values into the point-slope form:

• \(y - 0 = 2(x - 3)\)

This equation simplifies to:

• \(y = 2x - 6\)

By writing in this form, you can see how each piece relates to the components of the line: it starts at point (3,0) and rises 2 units for each 1 unit it moves right.

Standard Form of a Line

The standard form of a line is a uniform way to express linear equations and is given by:

• \(Ax + By = C\)

In this form, \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative. This form is particularly useful for equations not well-suited to the slope-intercept form, such as vertical lines.
For our line equation, starting from the simplified equation \(y = 2x - 6\), we rearrange the terms to put it into standard form:

• Subtract \(2x\) from both sides: \(-2x + y = -6\)
• Then multiply the entire equation by -1 to convert \(A\) into a positive integer: \(2x - y = 6\)

Now it's in the standard form \(2x - y = 6\), meeting all the criteria where \(A = 2\), \(B = -1\), and \(C = 6\). This method showcases the relationship between different forms of linear equations.