Question

write the equation of a line parallel to $y = -\frac{3}{4}x$ and passes through the point (4, -1).
$3x + 4y = 8$
$4x - 3y = 8$
$3x - 4y = 4$
$4x - y = 4$

Explanation:

Step1: Recall parallel line slope

Parallel lines have equal slopes. The given line \( y = -\frac{3}{4}x \) has slope \( m = -\frac{3}{4} \). So the new line also has slope \( -\frac{3}{4} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(4,-1) \) and \( m = -\frac{3}{4} \).
Substitute the values: \( y - (-1)=-\frac{3}{4}(x - 4) \)
Simplify: \( y + 1=-\frac{3}{4}x+3 \)
Multiply both sides by 4 to eliminate fraction: \( 4y + 4=-3x + 12 \)
Rearrange to standard form (\(Ax + By = C\)): \( 3x+4y=8 \)

Step3: Verify by plugging in the point

We can also check each option by plugging \( x = 4 \) and \( y=-1 \) into the equations.

• For \( 3x + 4y=8 \): \( 3(4)+4(-1)=12 - 4 = 8 \), which satisfies the equation.
• For \( 4x-3y = 8 \): \( 4(4)-3(-1)=16 + 3=19

eq8 \)

• For \( 3x - 4y=4 \): \( 3(4)-4(-1)=12 + 4 = 16

eq4 \)

• For \( 4x - y=4 \): \( 4(4)-(-1)=16 + 1=17

eq4 \)

Answer:

A. \( 3x + 4y = 8 \)