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Line [tex]$AB$[/tex] passes through [tex]$A(-3,0)$[/tex] and [tex]$B(-6,5)$[/tex]. What is the equation of the line that passes through the origin and is parallel to line [tex]$AB$[/tex]?

A. [tex]$5x - 3y = 0$[/tex]

B. [tex]$-x + 3y = 0$[/tex]

C. [tex]$-5x - 3y = 0$[/tex]

D. [tex]$3x + 5y = 0$[/tex]

E. [tex]$-3x + 5y = 0$[/tex]

Sagot :

GinnyAnsweridn
To find the equation of the line that passes through the origin and is parallel to the line passing through points [tex]\( A(-3, 0) \)[/tex] and [tex]\( B(-6, 5) \)[/tex], we need to follow these steps:

1. Find the slope of the line passing through points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
The slope [tex]\( m \)[/tex] of the line through the points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

For points [tex]\( A(-3, 0) \)[/tex] and [tex]\( B(-6, 5) \)[/tex]:
[tex]\[ m = \frac{5 - 0}{-6 - (-3)} = \frac{5}{-6 + 3} = \frac{5}{-3} = -\frac{5}{3} \][/tex]

2. Write the equation of the line passing through the origin with the same slope [tex]\( m \)[/tex]:
The general form of a line with slope [tex]\( m \)[/tex] passing through the origin [tex]\((0,0)\)[/tex] is:
[tex]\[ y = mx \][/tex]
Substituting [tex]\( m = -\frac{5}{3} \)[/tex]:
[tex]\[ y = -\frac{5}{3} x \][/tex]

3. Convert this equation to the standard form [tex]\( Ax + By = 0 \)[/tex]:
[tex]\[ y = -\frac{5}{3} x \quad \Rightarrow \quad 3y = -5x \quad \Rightarrow \quad 5x + 3y = 0 \][/tex]

To ensure it matches one of the choices given, rewrite it:
[tex]\[ -5x - 3y = 0 \][/tex]

4. Compare with the provided choices:
- A. [tex]\( 5x - 3y = 0 \)[/tex]
- B. [tex]\( -x + 3y = 0 \)[/tex]
- C. [tex]\( -5x - 3y = 0 \)[/tex]
- D. [tex]\( 3x + 5y = 0 \)[/tex]
- E. [tex]\( -3x + 5y = 0 \)[/tex]

Clearly, the equation [tex]\( -5x - 3y = 0 \)[/tex] matches choice [tex]\(\text{C.}\)[/tex]

So, the equation of the line that passes through the origin and is parallel to line [tex]\( AB \)[/tex] is:

C. [tex]\( -5x - 3y = 0 \)[/tex]
Chibabykalu16idn

Answer:

C. -5x - 3y = 0

Step-by-step explanation:

Given:

  • A(-3, 0) = A([tex]x_1, y_1[/tex])
  • B(-6, 5) = B([tex]x_2, y_2[/tex])

Firstly, find the slope of the line passing through points AB using the formula:

[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]

Substituting the values we get:

[tex]m = \frac{5 - 0}{-6 - (-3)}[/tex]

[tex]m = \frac{5}{-6 + 3}[/tex]

[tex]m = \frac{5}{-3}[/tex]

Write the equation of the line passing through the origin with the same slope:

[tex]y=mx[/tex]

[tex]\boxed{\text{\textbf{Note}: from previous, m = $\frac{5}{-3}$}}[/tex]

[tex]y=\frac{5}{-3}x[/tex]

[tex]y=\frac{5x}{-3}[/tex]

Cross-multiply

[tex]-3y=5x[/tex]

Move all terms to the LHS or RHS

[tex]-5x -3y=0[/tex]

[tex]5x + 3y = 0[/tex]

Therefore, the equation of the line that passes through the origin and is parallel to line AB is:

  • C. -5x - 3y = 0
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