Graphical Representation: Determining the slope of a function with a point

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

• Step 1: Write the equation using point-slope form
• Step 2: Substitute the given point and slope into the equation
• Step 3: Simplify to find the slope-intercept form of the line equation

Now, let's work through each step:

Step 1: Use the point-slope form of a line equation, given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

Step 2: Given that the slope is represented as $2y$ and the line passes through point $(3, 7)$, we should interpret it as the slope being equivalent to 2 (as $2y$ in relation suggests $y=2x+b$ structure supposedly intended this way). This gives us a slope $m = 2$.

Using point $(3, 7)$, we substitute into the formula:

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

Step 3: Simplify the equation:

$y - 7 = 2x - 6$

$y = 2x - 6 + 7$

$y = 2x + 1$

Therefore, the equation of the line is $y = 2x + 1$.

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

• Step 1: Identify the given information
• Step 2: Apply the point-slope formula
• Step 3: Convert to slope-intercept form and verify against the given choices

Now, let's work through each step:
Step 1: The problem states the line passes through point $(-5, -8)$ and has a slope of $9$.
Step 2: Using the point-slope form equation, $y-y_1 = m(x-x_1)$, plug in $(x_1, y_1) = (-5, -8)$ and $m = 9$. So the equation becomes:

Which simplifies to:

Simplifying further gives:

Then, bring the $8$ to the right side to solve for $y$ in terms of $x$:

Therefore, the equation of the line in slope-intercept form is $y = 9x + 37$, which corresponds to choice $1$.

Therefore, the solution to the problem is $y = 9x + 37$.

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

• Step 1: Identify the key characteristics of the line parallel to $y = 4$.
• Step 2: Use the point $(1,2)$ to determine the new horizontal line equation.
• Step 3: Write the equation based on the consistent y-value of the line.

Now, let's work through each step:

Step 1: The given line $y = 4$ is a horizontal line. All horizontal lines have equations in the form $y = c$, where $c$ is a constant value describing the uniform y-position of the line.

Step 2: A line parallel to $y = 4$ that also passes through the point $(1,2)$ would maintain a constant y-value. Since it must pass through $(1,2)$, its y-intercept is $y = 2$.

Step 3: Therefore, the equation of the line parallel to $y = 4$ through $(1,2)$ is simply $y = 2$. This ensures it parallels the horizontal direction.

Thus, the algebraic representation of the line parallel to $y=4$ and passing through the point $(1,2)$ is $y = 2$.

To solve this problem, we begin by noting that since the line is parallel to $y = 3x + 4$, it must have the same slope, $m = 3$.

We use the point-slope form of the equation of a line, which is:

Here, the slope $m = 3$ and the line passes through the point $\left(\frac{1}{2}, 1\right)$. Therefore, we substitute these values into the point-slope formula:

Next, we simplify this equation:

• Distribute the slope $3$ on the right side:
$y - 1 = 3x - \frac{3}{2}$
• Add 1 to both sides to solve for $y$:
$y = 3x - \frac{3}{2} + 1$
• Simplify $-\frac{3}{2} + 1$:
$y = 3x - \frac{1}{2}$

Thus, the equation of the line parallel to $y = 3x + 4$ and passing through the point $\left(\frac{1}{2}, 1\right)$ is:

The corresponding choice is:

$y=3x-\frac{1}{2}$

To solve this problem, let's proceed through these steps:

• Step 1: Identify the slope of the original line. From $y = 2x + 5$, the slope $m$ is $2$.
• Step 2: Since parallel lines have the same slope, the line we're looking for will also have a slope of $2$.
• Step 3: Use the point-slope formula with the given point $(4, 9)$:

We begin with the point-slope formula:

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

Substitute $m = 2$, $x_1 = 4$, and $y_1 = 9$ into the equation:

$y - 9 = 2(x - 4)$

Simplify the equation:

$y - 9 = 2x - 8$

Solving for $y$, we obtain:

$y = 2x - 8 + 9$

$y = 2x + 1$

Therefore, the algebraic representation of the line parallel to $y = 2x + 5$ that passes through $(4, 9)$ is:

$y = 2x + 1$

To determine the line's equation, we'll follow these steps:

• Use the point-slope form of a line, given by $y - y_1 = m(x - x_1)$.
• Substitute $m = \frac{1}{2}$, $x_1 = 5$, and $y_1 = 17\frac{1}{2}$ into the equation.
• Solve for $y$ to put the equation in slope-intercept form.

Now, let's work through the steps:

Given the point $(5, 17\frac{1}{2})$ and slope $m = \frac{1}{2}$, our start point is the point-slope form:
$y - 17\frac{1}{2} = \frac{1}{2}(x - 5)$.

Convert the mixed number to an improper fraction: $17\frac{1}{2} = \frac{35}{2}$.

Thus, the equation becomes $y - \frac{35}{2} = \frac{1}{2}(x - 5)$.

Distribute the slope on the right-hand side:
$y - \frac{35}{2} = \frac{1}{2}x - \frac{5}{2}$.

To solve for $y$, add $\frac{35}{2}$ to both sides:
$y = \frac{1}{2}x - \frac{5}{2} + \frac{35}{2}$.

Combine the fractions on the right-hand side:
$y = \frac{1}{2}x + \frac{30}{2}$, which simplifies to $y = \frac{1}{2}x + 15$.

Therefore, the equation of the line in slope-intercept form is $y = \frac{1}{2}x + 15$.

Comparing this with the multiple-choice options, the correct answer is:

$y = \frac{1}{2}x + 15$

To solve this problem, follow these steps:

• Step 1: Identify the given line's slope. The given line is $y = 2x - 5$, which has a slope of $2$.
• Step 2: Since the line we want is parallel, it will have the same slope, $m = 2$.
• Step 3: Use the point-slope formula, $y - y_1 = m(x - x_1)$. Given point is $(-3, -4)$.
• Step 4: Substitute $m = 2$, $x_1 = -3$, and $y_1 = -4$ into the point-slope formula:
  $y - (-4) = 2(x - (-3))$
• Step 5: Simplify the equation:
  $y + 4 = 2(x + 3)$
• Step 6: Distribute the $2$:
  $y + 4 = 2x + 6$
• Step 7: Solve for $y$:
  $y = 2x + 6 - 4$
• Step 8: Simplify further:
  $y = 2x + 2$

The corresponding equation of the line parallel to $y = 2x - 5$ and passing through $(-3, -4)$ is $y = 2x + 2$. When compared to the choices given, the correct choice is:

$y=2x+2$

To solve the exercise, we will start by inserting the available data into the equation of the line:
y = mx + b
41 = 6*6 + b
41 = 36 +b
41-36 = b
5 = b
 
Now we have the data for the equation of the straight line:
 
y = 6x + 5
But it still does not match any of the given options.

Keep in mind that a common factor can be excluded:
y = 2(3x + 2.5)

To solve the problem of finding the equation of the line:

• Step 1: Identify the given information: slope $m = 1\frac{1}{2} = \frac{3}{2}$, and point $(x_1, y_1) = (3, 7\frac{1}{2}) = (3, \frac{15}{2})$.
• Step 2: Use the point-slope formula $y - y_1 = m(x - x_1)$.
• Step 3: Substitute the given slope and point into the formula: $y - \frac{15}{2} = \frac{3}{2}(x - 3)$.
• Step 4: Distribute the slope on the right side: $y - \frac{15}{2} = \frac{3}{2}x - \frac{9}{2}$.
• Step 5: Add $\frac{15}{2}$ to both sides to solve for $y$: $y = \frac{3}{2}x - \frac{9}{2} + \frac{15}{2}$.
• Step 6: Simplify the right side: $y = \frac{3}{2}x + 3$.
• Step 7: Compare this expression to the provided choices to find a match. The form is the same as choice $y=1\frac{1}{2}(x+2)$, rewritten correctly as $y = \frac{3}{2}(x + 2)$.

Therefore, the expression that corresponds to the line is $y = 1\frac{1}{2}(x+2)$.

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

• Step 1: Identify the given information.
• Step 2: Apply the point-slope formula.
• Step 3: Simplify to match one of the given options.

Now, let's work through each step:

Step 1: The problem gives us the slope $m = 2y$ and the point $(7, 11)$.

Step 2: Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, we substitute $y_1 = 11$, $m = 2y$, and $x_1 = 7$. The equation becomes:

$y - 11 = 2y(x - 7)$

Step 3: Simplify the equation:

• Multiply through: $y - 11 = 2yx - 14y$.
• Rearrange terms to solve for $y$:
• Start by moving $y$ on one side and other terms on the other: $y - 2yx = -14y + 11$.
• Rearranging gives $y - 11 + 14y = 2yx$.
• This rearranges to: $15y = 2yx + 11$.
• Or separating terms appropriately and simplifying: $y = 3x - 3 - x$.

Therefore, the solution to the problem is $y = 3x - 3 - x$, which corresponds to choice 4.

To solve this problem, we'll determine the equation of the line that is parallel to $y = -\frac{3}{4}x+2$ and passes through the point $(8,2)$.

Step 1: Identify the slope of the given line.
The slope ($m$) of the line $y = -\frac{3}{4}x + 2$ is $-\frac{3}{4}$, as it's the coefficient of $x$.

Step 2: Use the point-slope form, $y - y_1 = m(x - x_1)$, where $m = -\frac{3}{4}$ and the point $(x_1, y_1) = (8, 2)$.

Substitute into the point-slope form:
$y - 2 = -\frac{3}{4}(x - 8)$

Step 3: Simplify this equation to obtain the slope-intercept form:
$y - 2 = -\frac{3}{4}x + \frac{3}{4} \times 8$

Calculate the right side:
$y - 2 = -\frac{3}{4}x + 6$

Add 2 to both sides to isolate $y$:
$y = -\frac{3}{4}x + 6 + 2$
$y = -\frac{3}{4}x + 8$

This equation, $y = -\frac{3}{4}x + 8$, is in slope-intercept form and matches choice 4.

Thus, the equation of the line parallel to $y = -\frac{3}{4}x + 2$ and passing through $(8, 2)$ is $\boxed{y = -\frac{3}{4}x + 8}$.