How to Write Linear Equations

How to Write Linear Equations?

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An equation presented in slope-intercept form is written as $y \ =\ mx \ + \ b$
Where $m$ is the slope of the line and $b$ is the $y$ -intercept. If you know the slope and the $y$ -intercept, you can use this equation to write a line equation.

Example

Find the equation of the line that passes through the points $(-1,0)$ and $(0,3)$ .

Writing Linear Equations

Solution:

We should find the $b$ -value and the $y$ -intercept, to do that, so we select two points on the line:

Writing Linear Equations2

Now, determine the slope between the $2$ points:

$m \ = \ \frac{y_2 \ - \ y_1}{x_2 \ - \ x_1} \ = \ \frac{0 \ - \ 3}{-1 \ - \ 0} \ = \ \frac{-3}{-1} \ = \ 3$

The $b$ -value can be found by just looking at the graph:

Writing Linear Equations3

$⇒ \ b \ = \ 3$

Now, we can use the equation $y \ =\ mx \ + \ b$ and write the line's equation:

$y \ =\ mx \ + \ b \ ⇒ \ y \ =\ 3x \ + \ 3$

In a lot of cases, the value of $b$ isn’t read so easily. In that case, or in cases where you are not sure if the line really crosses the $y$ -intercept at this particular point, $b$ can be calculated by solving the equation for $b$ , then substituting $x$ and $y$ with one of the two points.

The previous example can be used to explain this:

The two points are $(-1 \ , \ 0)$ , and $(0 \ , \ 3)$ . We determined the slope from these two points: $m \ = \ 3$
Now we have the equation: $y \ = \ 3x \ + \ b$
From that the equation can be solved for $b$ : $b \ = \ y \ - \ 3x$
Also, if we input the values from the second point $(0 \ , \ 3)$ , we have: $b \ = \ 3 \ - \ 3(0) \ = \ 3$
Finally, we can write the line equation: $y \ = \ 3x \ + 3$

Summary:

Writing linear equations:

Step1: Determine the slope, $m$ . This can be done by utilizing the slope formula to get the slope between two known points on the line.
Step2: Determine the $y$ -intercept. In order to do this, simply substitute the slope and the $(x \ , \ y)$ coordinates of a point on the line in the slope-intercept formula and then calculate $b$ .
Step3: Once you have $m$ and $b$ , you can enter them into the slope-intercept equation and find the line equation.

Free printable Worksheets

Exercises for Writing Linear Equations

1) Find the line equation of the line passing through $(10,9)(11,0)$ $\ \Rightarrow \$

2) Find the line equation of the line passing through $(8,7)(9,-5)$ $\ \Rightarrow \$

3) Find the line equation of the line passing through $(7,10)(8,-6)$ $\ \Rightarrow \$

4) Find the line equation of the line passing through $(9,5)(10,3)$ $\ \Rightarrow \$

5) Find the line equation of the line passing through $(10,2)(11,3)$ $\ \Rightarrow \$

6) Find the line equation of the line passing through $(1,4)(2,2)$ $\ \Rightarrow \$

7) Find the line equation of the line passing through $(7,3)(8,5)$ $\ \Rightarrow \$

8) Find the line equation of the line passing through $(2,9)(3,5)$ $\ \Rightarrow \$

9) Find the line equation of the line passing through $(3,6)(4,-4)$ $\ \Rightarrow \$

10) Find the line equation of the line passing through $(5,1)(6,-3)$ $\ \Rightarrow \$

Answers

1) Find the line equation of the line passing through

(10, 9) (11, 0)

$(10,9)(11,0)$

\Rightarrow m = \frac{0 - 9}{11 - 10} = - 9

$\ \Rightarrow \ \color{red}{m = \frac{0 \ - \ 9}{11 \ - \ 10} = -9}$

\Rightarrow y - 9 = m (x - 10)

$\ \Rightarrow \ \color{red}{y - 9= m(x - 10)}$

\Rightarrow y = - 9 x + 99

$\ \Rightarrow \ \color{red}{y = -9x \ + \ 99}$

2) Find the line equation of the line passing through

(8, 7) (9, - 5)

$(8,7)(9,-5)$

\Rightarrow m = \frac{- 5 - 7}{9 - 8} = - 12

$\ \Rightarrow \ \color{red}{m = \frac{-5 \ - \ 7}{9 \ - \ 8} = -12}$

\Rightarrow y - 7 = m (x - 8)

$\ \Rightarrow \ \color{red}{y - 7= m(x - 8)}$

\Rightarrow y = - 12 x + 103

$\ \Rightarrow \ \color{red}{y = -12x \ + \ 103}$

3) Find the line equation of the line passing through

(7, 10) (8, - 6)

$(7,10)(8,-6)$

\Rightarrow m = \frac{- 6 - 10}{8 - 7} = - 16

$\ \Rightarrow \ \color{red}{m = \frac{-6 \ - \ 10}{8 \ - \ 7} = -16}$

\Rightarrow y - 10 = m (x - 7)

$\ \Rightarrow \ \color{red}{y - 10= m(x - 7)}$

\Rightarrow y = - 16 x + 122

$\ \Rightarrow \ \color{red}{y = -16x \ + \ 122}$

4) Find the line equation of the line passing through

(9, 5) (10, 3)

$(9,5)(10,3)$

\Rightarrow m = \frac{3 - 5}{10 - 9} = - 2

$\ \Rightarrow \ \color{red}{m = \frac{3 \ - \ 5}{10 \ - \ 9} = -2}$

\Rightarrow y - 5 = m (x - 9)

$\ \Rightarrow \ \color{red}{y - 5= m(x - 9)}$

\Rightarrow y = - 2 x + 23

$\ \Rightarrow \ \color{red}{y = -2x \ + \ 23}$

5) Find the line equation of the line passing through

(10, 2) (11, 3)

$(10,2)(11,3)$

\Rightarrow m = \frac{3 - 2}{11 - 10} = 1

$\ \Rightarrow \ \color{red}{m = \frac{3 \ - \ 2}{11 \ - \ 10} = 1}$

\Rightarrow y - 2 = m (x - 10)

$\ \Rightarrow \ \color{red}{y - 2= m(x - 10)}$

\Rightarrow y = x - 8

$\ \Rightarrow \ \color{red}{y = x \ - \ 8}$

6) Find the line equation of the line passing through

(1, 4) (2, 2)

$(1,4)(2,2)$

\Rightarrow m = \frac{2 - 4}{2 - 1} = - 2

$\ \Rightarrow \ \color{red}{m = \frac{2 \ - \ 4}{2 \ - \ 1} = -2}$

\Rightarrow y - 4 = m (x - 1)

$\ \Rightarrow \ \color{red}{y - 4= m(x - 1)}$

\Rightarrow y = - 2 x + 6

$\ \Rightarrow \ \color{red}{y = -2x \ + \ 6}$

7) Find the line equation of the line passing through

(7, 3) (8, 5)

$(7,3)(8,5)$

\Rightarrow m = \frac{5 - 3}{8 - 7} = 2

$\ \Rightarrow \ \color{red}{m = \frac{5 \ - \ 3}{8 \ - \ 7} = 2}$

\Rightarrow y - 3 = m (x - 7)

$\ \Rightarrow \ \color{red}{y - 3= m(x - 7)}$

\Rightarrow y = 2 x - 11

$\ \Rightarrow \ \color{red}{y = 2x \ - \ 11}$

8) Find the line equation of the line passing through

(2, 9) (3, 5)

$(2,9)(3,5)$

\Rightarrow m = \frac{5 - 9}{3 - 2} = - 4

$\ \Rightarrow \ \color{red}{m = \frac{5 \ - \ 9}{3 \ - \ 2} = -4}$

\Rightarrow y - 9 = m (x - 2)

$\ \Rightarrow \ \color{red}{y - 9= m(x - 2)}$

\Rightarrow y = - 4 x + 17

$\ \Rightarrow \ \color{red}{y = -4x \ + \ 17}$

9) Find the line equation of the line passing through

(3, 6) (4, - 4)

$(3,6)(4,-4)$

\Rightarrow m = \frac{- 4 - 6}{4 - 3} = - 10

$\ \Rightarrow \ \color{red}{m = \frac{-4 \ - \ 6}{4 \ - \ 3} = -10}$

\Rightarrow y - 6 = m (x - 3)

$\ \Rightarrow \ \color{red}{y - 6= m(x - 3)}$

\Rightarrow y = - 10 x + 36

$\ \Rightarrow \ \color{red}{y = -10x \ + \ 36}$

10) Find the line equation of the line passing through

(5, 1) (6, - 3)

$(5,1)(6,-3)$

\Rightarrow m = \frac{- 3 - 1}{6 - 5} = - 4

$\ \Rightarrow \ \color{red}{m = \frac{-3 \ - \ 1}{6 \ - \ 5} = -4}$

\Rightarrow y - 1 = m (x - 5)

$\ \Rightarrow \ \color{red}{y - 1= m(x - 5)}$

\Rightarrow y = - 4 x + 21

$\ \Rightarrow \ \color{red}{y = -4x \ + \ 21}$

How to Write Linear Equations