Questions: Question 15 of 20 Write an equation of the line passing through the given point and satisfying the given condition. Give the equation (a) in slope-intercept form and (b) in standard form. (8,1); parallel to 2x-y=9 (a) Write the equation of the line in slope-intercept form. (Simplify your answer. Use integers or fractions for any numbers in the equation.) (b) Write the equation of the line in standard form. (Simplify your answer. Use integers or fractions for any numbers in the equation.)

Solution Steps

To solve this problem, we need to find the equation of a line that passes through a given point and is parallel to a given line.

1. Identify the slope of the given line: The given line is in the form \(2x - y = 9\). We can rewrite this in slope-intercept form \(y = mx + b\) to find the slope \(m\).
2. Use the point-slope form: With the slope from step 1 and the given point \((8, 1)\), use the point-slope form of the equation \(y - y_1 = m(x - x_1)\) to find the equation of the line.
3. Convert to slope-intercept form: Simplify the equation from step 2 to get it in the form \(y = mx + b\).
4. Convert to standard form: Rearrange the equation from step 3 to get it in the form \(Ax + By = C\).

The given line is represented by the equation \(2x - y = 9\). To find its slope, we can rewrite it in slope-intercept form \(y = mx + b\): \[ y = 2x - 9 \] From this, we see that the slope \(m\) of the given line is \(2\).

We need to find the equation of the line that is parallel to the given line and passes through the point \((8, 1)\). Using the point-slope form of the equation: \[ y - y_1 = m(x - x_1) \] Substituting \(m = 2\), \(x_1 = 8\), and \(y_1 = 1\): \[ y - 1 = 2(x - 8) \]

Now, we simplify the equation to get it in slope-intercept form: \[ y - 1 = 2x - 16 \] \[ y = 2x - 15 \]

Next, we convert the slope-intercept form \(y = 2x - 15\) to standard form \(Ax + By = C\): \[ 2x - y = 15 \]

Final Answer