Linear Equation Practice Problems | Equation of Straight Line

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).

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.

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:

$y=-1\times x+5$

$y=-x+5$