Equation of a Straight Line - Examples, Exercises and Solutions

In the first step, we'll find the slope using the formula:

$m=\frac{y_2-y_1}{x_2-x_1}$

We'll substitute according to the given points:

$m=\frac{12-(-6)}{4-(-2)}$

$m=\frac{18}{6}=3$

Now we'll choose the point (4,12) and use the formula:

$y=mx+b$

$12=3\times4+b$

$12=12+b$

$b=0$

We'll substitute the data into the formula to find the equation of the line:

$y=3x$

In the first step, we'll find the slope using the formula:

$m=\frac{y_2-y_1}{x_2-x_1}$

We'll substitute according to the given points:

$m=\frac{1-2}{\frac{1}{3}-(-\frac{1}{3})}$

$m=\frac{-1}{\frac{2}{3}}=-\frac{3}{2}$

Now we'll choose point $(\frac{1}{3},1)$ and use the formula:

$y=mx+b$

$1=-\frac{3}{2}\times\frac{1}{3}+b$

$1=-\frac{1}{2}+b$

$b=1\frac{1}{2}$

We'll substitute the given data into the formula to find the equation of the line:

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

To find the equation of the line passing through the points $(9, 10)$ and $(99, 100)$, follow these steps:

• Step 1: Determine the slope, $m$, of the line.
• Step 2: Use one point and the calculated slope to find the equation of the line.
• Step 3: Simplify to find the line's equation in slope-intercept form.

Step 1: Calculate the slope $m$ using the formula:
$m = \frac{100 - 10}{99 - 9} = \frac{90}{90} = 1$.

Step 2: Now, use the point $(9, 10)$ and the point-slope form:
$y - 10 = 1 \cdot (x - 9)$.

Step 3: Simplify this equation:
$y - 10 = x - 9$
$y = x + 1$.

Thus, the equation of the line is $y = x + 1$, matching answer choice (2).

First, we will use the formula to find the slope of the straight line:

We replace the data and solve:

$\frac{(0-4.5)}{(5-0.5)}=\frac{-4.5}{4.5}=-1$

Now, we know that the slope is $-1$

We replace one of the points in the formula of the line equation:

$y=mx+b$

$(5,0)$

$0=-1\times5+b$

 $0=-5+b$

$b=5$

Now we have the data to complete the equation:

$y=-1\times x+5$

$y=-x+5$

Let's solve the problem to find the equation of the line.

To determine the equation of the line, we first need to calculate the slope $m$ of the line passing through the points $(15, 36)$ and $(5, 16)$. The formula for the slope is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the given points into the formula:

$m = \frac{16 - 36}{5 - 15} = \frac{-20}{-10} = 2$

Thus, the slope $m$ is 2.

With the slope and one of the points, we can use the point-slope form of the line equation:

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

We'll use the point $(5, 16)$:

$y - 16 = 2(x - 5)$

Expanding the equation, we get:

$y - 16 = 2x - 10$

Add 16 to both sides to solve for $y$:

$y = 2x - 10 + 16$

$y = 2x + 6$

Therefore, the equation of the line is $y = 2x + 6$.

To find the equation of the line passing through the points $(12,40)$ and $(2,10)$, follow these steps:

• Calculate the slope $m$.
• Use the point-slope form to find the equation.

Step 1: Calculate the slope $m$.
Using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where $(x_1, y_1) = (2, 10)$ and $(x_2, y_2) = (12, 40)$, we find:

$m = \frac{40 - 10}{12 - 2} = \frac{30}{10} = 3$

Step 2: Use the point-slope form $y - y_1 = m(x - x_1)$.
We use one of the points, say $(2,10)$, and the calculated slope $m = 3$ to write:

$y - 10 = 3(x - 2)$

Simplify the equation:
$y - 10 = 3x - 6$

Add 10 to both sides:
$y = 3x - 6 + 10$

$y = 3x + 4$

The line equation is $y = 3x + 4$. Comparing to the given correct answer and available choices, we note:

Therefore, the expression $y - 4 = 3x$ is equivalent to $y = 3x + 4$, matching Choice 1, considered correctly due to specific transformation and option alignment.

This equivalent manipulation keeps the original expression correct while aligning with the structural prompt.

Thus, the equation of the line is therefore $y - 4 = 3x$.

To find the equation of the line passing through the points $(2,8)$ and $(6,1)$, follow the steps below:

• Step 1: Calculate the slope $m$.
  The formula for the slope $m$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 8}{6 - 2} = \frac{-7}{4}$

• Step 2: Use the point-slope form to write the equation of the line.
  The point-slope form of a line is given by:

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

Using the point $(2,8)$:

$y - 8 = -\frac{7}{4}(x - 2)$

• Step 3: Simplify to the slope-intercept form.
  Distribute the slope and rearrange to find $y$:

$y - 8 = -\frac{7}{4}x + \frac{7}{2}$

Add 8 to both sides to solve for $y$:

$y = -\frac{7}{4}x + \frac{7}{2} + 8$

Convert 8 to a fraction with a denominator of 2:

$y = -\frac{7}{4}x + \frac{7}{2} + \frac{16}{2}$

Simplify the addition:

$y = -\frac{7}{4}x + \frac{23}{2}$

To convert $23/2$ into mixed number form: $\frac{23}{2} = 11 \frac{1}{2}$

Thus, the equation in slope-intercept form is: $y = -\frac{7}{4}x + 11 \frac{1}{2}$

Therefore, the equation of the line passing through these points is $y = -1\frac{3}{4}x + 11\frac{1}{2}$, which matches the correct choice in the multiple-choice answers.

To solve this problem, we will follow these steps:

• Step 1: Calculate the slope of the line passing through the points $(5, -11)$ and $(1, 9)$.

• Step 2: Use the calculated slope and one of the points to determine the equation of the line in point-slope form.

• Step 3: Convert the equation to standard form and verify with choices.

Step 1: Calculate the Slope
The slope $m$ is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - (-11)}{1 - 5} = \frac{9 + 11}{1 - 5} = \frac{20}{-4} = -5$

Step 2: Use Point-Slope Form
We choose point $(1, 9)$ to use in the point-slope formula:

$y - y_1 = m(x - x_1)$
$y - 9 = -5(x - 1)$

Step 3: Simplify to Standard Form
Expand and rearrange the equation:

$y - 9 = -5x + 5$

Bring all terms to one side:

$y + 5x = 14$

The linear equation in standard form is $y + 5x = 14$, which matches choice 4.

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 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 $y = x + 8$.

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 find which line forms a greater angle with the x-axis, we use the formula $\theta = \tan^{-1}(m)$. This gives us the angles formed by lines with slopes $m_1 = -3$ and $m_2 = -6$.

We first calculate the inverse tangent for both slopes:

• For $m_1 = -3$, the angle is $\theta_1 = \tan^{-1}(-3)$.
• For $m_2 = -6$, the angle is $\theta_2 = \tan^{-1}(-6)$.

The greater the absolute value of the negative slope, the closer the line is to the vertical, thus forming a greater angle with the x-axis.

Comparing $|-3|$ and $|-6|$, we see that $-6$ has a greater absolute value, indicating a steeper angle. Hence, although $-6$ is steeper, it forms a smaller angle with the x-axis because we're considering angles formed in the positive x-direction (i.e., above the x-axis).

Therefore, the line whose slope is $-3$ forms the greater angle with the x-axis.

The line whose slope is -3 forms the greater angle.

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 for which line forms a larger angle with the x-axis, we'll proceed by calculating the arctangent of each slope:

• First, calculate the angle for the line with slope $m_1 = 2$:

$\theta_1 = \tan^{-1}(2)$

• Next, calculate the angle for the line with slope $m_2 = \frac{1}{2}$:

$\theta_2 = \tan^{-1}\left(\frac{1}{2}\right)$

Comparing these two results:

• The function $\tan^{-1}(x)$ is increasing, meaning as the value of $x$ increases, so does the angle $\theta$. Therefore, since $2 > \frac{1}{2}$, it follows that $\theta_1 > \theta_2$.

Therefore, the straight line with a slope of 2 forms the largest angle with the x-axis.

We will use the formula:

$m=\tan\alpha$

Let's check the slope of minus 6:

$-6=\tan\alpha$

$\tan^{-1}(-6)=\alpha$

$-80.53=\alpha$

$180-80.53=$

$99.47=\alpha_1$

Let's check the slope of one-half:

$\frac{1}{2}=\tan\alpha$

$\tan^{-1}(\frac{1}{2})=\alpha$

$26.56=\alpha_2$

\alpha_1 > \alpha_2