Using points and equations

Question

Find the point on the line 5𝑥 + 3𝑦 - 4 = 0 which is closest to the point (-5,-4).

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Answer 1

This is an Algebra 2 problem.

1) determine slope of the given equation

2) determine slope of perpendicular to line in 1

3) determine equation of perpendicular passing through given point

3) determine point of intersection of given equation and perpendicular.

You post many problems. Is this due to a lack of understanding or to a lack of effort?

Answer 2

# Problem: Find the Point on the Line Closest to \((-5, -4)\)

To find the point on the line \( 5x + 3y - 4 = 0 \) that is closest to the point \((-5, -4)\), we'll follow these steps:

# Step 1: Find the Slope of the Given Line

The equation of the line is \( 5x + 3y - 4 = 0 \). Rewriting in slope-intercept form:

\[

3y = -5x + 4 \quad \Rightarrow \quad y = -\frac{5}{3}x + \frac{4}{3}

\]

The slope \( m_1 \) of the line is \( -\frac{5}{3} \).

#### Step 2: Find the Slope of the Perpendicular Line

The slope of the line perpendicular to this one, \( m_2 \), is the negative reciprocal of \( m_1 \):

\[

m_2 = \frac{3}{5}

\]

#### Step 3: Write the Equation of the Perpendicular Line

The perpendicular line passes through the point \((-5, -4)\) and has slope \( \frac{3}{5} \). Using the point-slope form:

\[

y + 4 = \frac{3}{5}(x + 5)

\]

Expanding:

\[

y + 4 = \frac{3}{5}x + 3 \quad \Rightarrow \quad y = \frac{3}{5}x - 1

\]

#### Step 4: Find the Intersection of the Two Lines

To find the intersection point of \( y = \frac{3}{5}x - 1 \) and \( 5x + 3y - 4 = 0 \):

Substitute \( y = \frac{3}{5}x - 1 \) into \( 5x + 3y - 4 = 0 \):

\[

5x + 3\left(\frac{3}{5}x - 1\right) - 4 = 0

\]

Simplifying:

\[

5x + \frac{9}{5}x - 7 = 0 \quad \Rightarrow \quad 34x = 35 \quad \Rightarrow \quad x = \frac{35}{34}

\]

#### Step 5: Find the Corresponding \( y \) Value

Substitute \( x = \frac{35}{34} \) into \( y = \frac{3}{5}x - 1 \):

\[

y = \frac{105}{170} - 1 = \frac{-65}{170} = -\frac{13}{34}

\]

### Final Answer

The point on the line \( 5x + 3y - 4 = 0 \) that is closest to the point \((-5, -4)\) is:

\[

\left(\frac{35}{34}, -\frac{13}{34}\right)

\]

Answer 3

Since this is posted as a calculus problem here is how you can find the point using optimization.

If you solve the given equation for y:

y = (-5/3)x + 4/3

That means every point on the line will have coordinates (x , [-5/3]x + 4/3).

Use the distance formula to create a function depending on x that gives the distance from a point on the line to the point (-5,-4):

D² = (x + 5)² + ((-5/3)x + 4/3 +12/3)²

D² = (x+5)² +(-5x/3 + 16/3)²

Take the derivative with respect to x:

2D dD/dx = 2(x+5) +2(-5x/3 + 16/3)(-5/3)

D' = [2(x+5) + 2(25x/9 - 80/9)]/2D

D' =[ (x+5) + (25x/9 - 80/9]/D

This is equal to zero when the numerator equals 0.

x + 5 + 25x/9 - 80/9 = 0

9x +45 +25x -80 = 0

34x - 35 = 0

x = 35/34

Here you can apply either the 1st or 2nd derivative test to show that this x value gives the minimum value for D.

When x = 35/34, y = (-5/3)(35/34) + 4/3 = -13/34.

So P(35/34, -13/34) is the point on the line closest to the point (-5,-4).

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