Equation of a Straight Line: Point-Slope Form of a Line

To solve the problem of determining the equation of the line parallel to $x + 3y = 4x + 9$ and passing through point $(6, 14)$, we will follow these steps:

• Step 1: Rearrange the given line to find its slope.
• Step 2: Use the slope and the given point to apply the point-slope form of a line.
• Step 3: Convert the equation into the slope-intercept form for clarity.

Let's perform each step:

Step 1: First, the equation $x + 3y = 4x + 9$ simplifies to find the slope. Subtract $x$ from both sides to get $3y = 3x + 9$. By dividing everything by 3, this gives $y = x + 3$. Therefore, the slope, $m$, is 1.

Step 2: Given that parallel lines share the same slope, the slope of the desired line is also 1. Applying the point-slope form: $y - 14 = 1(x - 6)$.

Step 3: Simplifying this equation yields $y - 14 = x - 6$. Further rearranging gives $y = x + 8$.

Thus, the equation of the line parallel to the given line and passing through $(6, 14)$ is $y = x + 8$.

First, write out the line equations:

$-4+y=5x-6$

$y=5x+4-6$

$y=5x-2$

From here we can determine the slope:

$m=5$

We'll use the formula:

$y=mx+b$

We'll use the point $(-2,-13)$:

$-13=5\times-2+b$

$-13=-10+b$

$-3=b$

Finally, substitute our data back into the formula:

$y=5x+(-3)$

$y=5x-3$

To solve this problem, we'll follow these steps:

• Step 1: Determine the slope of the given line $5y - 5x = 15$.
• Step 2: Use this slope and the point $(-5, 12)$ in the point-slope form.
• Step 3: Rearrange the equation to find the required line's equation.

Let's work through each step:

Step 1: Determine the slope of the given line.
First, convert the equation $5y - 5x = 15$ to slope-intercept form ($y = mx + b$):

Divide every term by 5 to simplify:
$5y = 5x + 15$
$y = x + 3$

The slope ($m$) of this line is $1$.

Step 2: Use the point-slope form with the given point and slope.
We have a point $(-5, 12)$ and a slope $m = 1$. The point-slope form is:

$y - y_1 = m(x - x_1)$
Substitute $y_1 = 12$, $x_1 = -5$, and $m = 1$:
$y - 12 = 1(x + 5)$

Simplify the equation:
$y - 12 = x + 5$

Step 3: Rearrange to get the solution.
Rearrange the equation:
$y = x + 5 + 12$
$y = x + 17$

We want to express this in the form of one of the given choices:
$y - x = 17$.

Therefore, the solution to the problem is $y - x = 17$.

This matches the final answer choice given in the problem, confirming our solution is correct.

To solve for the equation of the line parallel to $2(y+1)=16x$ that passes through the point $(0,1)$, follow these steps:

• Step 1: Convert the given line's equation to slope-intercept form. Start with $2(y+1) = 16x$.
• Step 2: Distribute and simplify to get $2y + 2 = 16x$, which rearranges to $2y = 16x - 2$.
• Step 3: Solve for $y$ by dividing everything by 2: $y = 8x - 1$. Hence, the slope $m$ is 8.
• Step 4: Use the slope of 8 and the point $(0,1)$ in the point-slope form of a line equation:
  $\quad y - y_1 = m(x - x_1)$
• Step 5: Substitute the point $(0,1)$ and the slope $8$:
  $\quad y - 1 = 8(x - 0)$
• Step 6: Simplify the expression
  $\quad y - 1 = 8x$
• Step 7: Rewrite the equation to achieve the desired form by moving 1 to the other side:
  $\quad y = 8x + 1$
• Step 8: To match the expected form, rewrite
  $y - 8x = 1$

Therefore, the equation of the line that meets the criteria is given by $y - 8x = 1$, corresponding to choice 3.

To solve this problem, follow these steps:

Step 1: Identify the slope of the given line.

The given line is $y = -\frac{1}{5}x + 3$. Here, the slope ($m$) is $-\frac{1}{5}$.

Step 2: Determine the slope of the perpendicular line.

For lines to be perpendicular, the product of their slopes must equal $-1$. Hence, if $m_1 = -\frac{1}{5}$ is the slope of the given line, the slope ($m_2$) of the line perpendicular to it can be found using:

$m_1 \times m_2 = -1$, which implies:

$-\frac{1}{5} \times m_2 = -1$.

Solve for $m_2$ to get:

$m_2 = 5$.

Step 3: Use the point-slope form to find the equation of the line.

The line passes through point $(2, 9)$, and we have determined $m_2 = 5$.

Using the point-slope form $y - y_1 = m(x - x_1)$, substitute $m = 5$, $x_1 = 2$, $y_1 = 9$:

$y - 9 = 5(x - 2)$.

Step 4: Simplify the equation.

Distribute the 5:

$y - 9 = 5x - 10$.

Add 9 to both sides to solve for $y$:

$y = 5x - 1$.

Therefore, the equation of the line we are looking for is $\boxed{y = 5x - 1}$.

To solve this problem, we'll follow these steps:

• Step 1: Rearrange the given line equation to standard slope-intercept form.
• Step 2: Determine the slope of the perpendicular line.
• Step 3: Use the point-slope form to find the equation of the desired line.

Now, let's work through each step:

Step 1: The given line equation is $3y = 4x - y$. First, simplify this equation:

$3y = 4x - y$

Add $y$ to both sides to consolidate $y$:

$3y + y = 4x$

$4y = 4x$

Divide both sides by 4:

$y = x$

The slope $m$ of this line is 1.

Step 2: Since the line we want is perpendicular to this line, the slope $m_1$ of the desired line should satisfy:

$m \times m_1 = -1$

Given $m = 1$, we have:

$1 \times m_1 = -1$

So, $m_1 = -1$.

Step 3: Use the point-slope form of a line $y - y_1 = m(x - x_1)$ with point $(-3, -15)$ and slope $-1$:

$y - (-15) = -1(x - (-3))$

Simplify the equation:

$y + 15 = -1(x + 3)$

Expand:

$y + 15 = -x - 3$

Subtract 15 from both sides:

$y = -x - 3 - 15$

$y = -x - 18$

Rearrange to the standard form:

$y + x = -18$

The equation of the line that passes through $(-3, -15)$ and is perpendicular to $3y = 4x - y$ is:

$y + x = -18$

To solve this problem, we'll follow these steps:

• Simplify the given equation to find its slope.
• Determine the slope for the perpendicular line using the negative reciprocal.
• Write the equation of the line through the origin with this slope.

Let's work through each step:
Step 1: The original line is given by the equation $y - 2x = 2y$. Simplifying, we subtract $y$ from both sides to get $-2x = y$ or, equivalently, $y = -2x$. Here, the slope $m_1 = -2$.

Step 2: For the line to be perpendicular, its slope $m_2$ must satisfy $m_1 \cdot m_2 = -1$. Thus, we have $-2 \cdot m_2 = -1$, yielding $m_2 = \frac{1}{2}$.

Step 3: The equation of the line with this slope that passes through the origin is of the form $y = mx$. Substituting the slope $\frac{1}{2}$, we have $y = \frac{1}{2}x$.

Therefore, the function representing the straight line through the origin and perpendicular to the given line is $y = \frac{1}{2}x$, which corresponds to choice 1.

To solve this problem, we need to determine the equation of a line passing through the point $(7, 2)$ and perpendicular to the given line $3y = x + 12$. We can achieve this by following a systematic approach:

• Step 1: Identify the slope of the given line

We start by rewriting the equation of the given line $3y = x + 12$ in slope-intercept form, $y = mx + b$, where $m$ is the slope. To do this, we divide each side of the equation by 3:

$y = \frac{1}{3}x + 4$

Thus, the slope $m_1$ of the given line is $\frac{1}{3}$.

• Step 2: Determine the slope of the perpendicular line

Two lines are perpendicular if the product of their slopes is $-1$. Therefore, if the slope of the line we are looking for is $m_2$, then:

$m_1 \times m_2 = -1$

Substituting the value of $m_1$:

$\frac{1}{3} \times m_2 = -1$

Solving for $m_2$, we find:

$m_2 = -3$

• Step 3: Use the point-slope form to find the equation of the perpendicular line

Now that we know the slope of the perpendicular line is $-3$ and it passes through the point $(7, 2)$, we can use the point-slope form of a line:

$y - y_1 = m(x - x_1)$

Here, $m = -3$ and the point is $(x_1, y_1) = (7, 2)$, so:

$y - 2 = -3(x - 7)$

Expanding this equation to solve for $y$, we get:

$y - 2 = -3x + 21$

Adding 2 to both sides to isolate $y$, we have:

$y = -3x + 23$

Thus, the equation of the line that passes through the point $(7, 2)$ and is perpendicular to the line $3y = x + 12$ is $y = -3x + 23$.

The correct answer to this problem is therefore $y = -3x + 23$.