Question

1. write an equation that represents the line that is perpendicular to $4x - 3y + 5 = 0$ and passes through the point $(-3,7)$. express your answer in slope - point form. 3 points
2. a line a passes through a points $p(2,3)$ and $q(6,9)$. line b passes through the points $a(x,2)$ and $b(5,5)$. line b is parallel to line a. determine the value of $x$. 3 points

Explanation:

Problem 3

Step1: Find slope of given line

Rewrite \(4x - 3y + 5 = 0\) in slope - intercept form (\(y=mx + b\)).
\(-3y=-4x - 5\), so \(y=\frac{4}{3}x+\frac{5}{3}\). The slope (\(m_1\)) of the given line is \(\frac{4}{3}\).

Step2: Find slope of perpendicular line

If two lines are perpendicular, the product of their slopes is \(- 1\) (\(m_1\times m_2=-1\)). Let \(m_2\) be the slope of the perpendicular line.
\(\frac{4}{3}\times m_2=-1\), so \(m_2 =-\frac{3}{4}\).

Step3: Use slope - point form

The slope - point form of a line is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(-3,7)\) and \(m =-\frac{3}{4}\).
Substitute the values: \(y - 7=-\frac{3}{4}(x + 3)\) (since \(x-(-3)=x + 3\)).

Problem 4

Step1: Find slope of Line A

Line A passes through \(P(2,3)\) and \(Q(6,9)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For Line A, \(m_A=\frac{9 - 3}{6 - 2}=\frac{6}{4}=\frac{3}{2}\).

Step2: Find slope of Line B

Line B passes through \(A(x,2)\) and \(B(5,5)\). Using the slope formula, \(m_B=\frac{5 - 2}{5 - x}=\frac{3}{5 - x}\).

Step3: Set slopes equal (parallel lines)

Since Line A and Line B are parallel, their slopes are equal (\(m_A=m_B\)).
\(\frac{3}{2}=\frac{3}{5 - x}\). Cross - multiply: \(3(5 - x)=3\times2\).
Divide both sides by 3: \(5 - x = 2\).

Step4: Solve for \(x\)

Subtract 5 from both sides: \(-x=2 - 5=-3\). Multiply both sides by \(- 1\): \(x = 3\).

Problem 3 Answer: \(y - 7=-\frac{3}{4}(x + 3)\)

Problem 4 Answer: \(x = 3\)

Answer:

Step1: Find slope of Line A

Line A passes through \(P(2,3)\) and \(Q(6,9)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For Line A, \(m_A=\frac{9 - 3}{6 - 2}=\frac{6}{4}=\frac{3}{2}\).

Step2: Find slope of Line B

Line B passes through \(A(x,2)\) and \(B(5,5)\). Using the slope formula, \(m_B=\frac{5 - 2}{5 - x}=\frac{3}{5 - x}\).

Step3: Set slopes equal (parallel lines)

Since Line A and Line B are parallel, their slopes are equal (\(m_A=m_B\)).
\(\frac{3}{2}=\frac{3}{5 - x}\). Cross - multiply: \(3(5 - x)=3\times2\).
Divide both sides by 3: \(5 - x = 2\).

Step4: Solve for \(x\)

Subtract 5 from both sides: \(-x=2 - 5=-3\). Multiply both sides by \(- 1\): \(x = 3\).

Problem 3 Answer: \(y - 7=-\frac{3}{4}(x + 3)\)

Problem 4 Answer: \(x = 3\)