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 \((5,1)(6,-3)\)\( \ \Rightarrow \ \)

 
1) Find the line equation of the line passing through \((10,9)(11,0)\)\( \ \Rightarrow \ \color{red}{m = \frac{0 \ - \ 9}{11 \ - \ 10} = -9}\)\( \ \Rightarrow \ \color{red}{y - 9= m(x - 10)} \)\(\ \Rightarrow \ \color{red}{y = -9x \ + \ 99} \)
2) Find the line equation of the line passing through \((8,7)(9,-5)\)\( \ \Rightarrow \ \color{red}{m = \frac{-5 \ - \ 7}{9 \ - \ 8} = -12}\)\( \ \Rightarrow \ \color{red}{y - 7= m(x - 8)} \)\(\ \Rightarrow \ \color{red}{y = -12x \ + \ 103} \)
3) Find the line equation of the line passing through \((7,10)(8,-6)\)\( \ \Rightarrow \ \color{red}{m = \frac{-6 \ - \ 10}{8 \ - \ 7} = -16}\)\( \ \Rightarrow \ \color{red}{y - 10= m(x - 7)} \)\(\ \Rightarrow \ \color{red}{y = -16x \ + \ 122} \)
4) Find the line equation of the line passing through \((9,5)(10,3)\)\( \ \Rightarrow \ \color{red}{m = \frac{3 \ - \ 5}{10 \ - \ 9} = -2}\)\( \ \Rightarrow \ \color{red}{y - 5= m(x - 9)} \)\(\ \Rightarrow \ \color{red}{y = -2x \ + \ 23} \)
5) Find the line equation of the line passing through \((10,2)(11,3)\)\( \ \Rightarrow \ \color{red}{m = \frac{3 \ - \ 2}{11 \ - \ 10} = 1}\)\( \ \Rightarrow \ \color{red}{y - 2= m(x - 10)} \)\(\ \Rightarrow \ \color{red}{y = x \ - \ 8} \)
6) Find the line equation of the line passing through \((1,4)(2,2)\)\( \ \Rightarrow \ \color{red}{m = \frac{2 \ - \ 4}{2 \ - \ 1} = -2}\)\( \ \Rightarrow \ \color{red}{y - 4= m(x - 1)} \)\(\ \Rightarrow \ \color{red}{y = -2x \ + \ 6} \)
7) Find the line equation of the line passing through \((7,3)(8,5)\)\( \ \Rightarrow \ \color{red}{m = \frac{5 \ - \ 3}{8 \ - \ 7} = 2}\)\( \ \Rightarrow \ \color{red}{y - 3= m(x - 7)} \)\(\ \Rightarrow \ \color{red}{y = 2x \ - \ 11} \)
8) Find the line equation of the line passing through \((2,9)(3,5)\)\( \ \Rightarrow \ \color{red}{m = \frac{5 \ - \ 9}{3 \ - \ 2} = -4}\)\( \ \Rightarrow \ \color{red}{y - 9= m(x - 2)} \)\(\ \Rightarrow \ \color{red}{y = -4x \ + \ 17} \)
9) Find the line equation of the line passing through \((3,6)(4,-4)\)\( \ \Rightarrow \ \color{red}{m = \frac{-4 \ - \ 6}{4 \ - \ 3} = -10}\)\( \ \Rightarrow \ \color{red}{y - 6= m(x - 3)} \)\(\ \Rightarrow \ \color{red}{y = -10x \ + \ 36} \)
10) Find the line equation of the line passing through \((5,1)(6,-3)\)\( \ \Rightarrow \ \color{red}{m = \frac{-3 \ - \ 1}{6 \ - \ 5} = -4}\)\( \ \Rightarrow \ \color{red}{y - 1= m(x - 5)} \)\(\ \Rightarrow \ \color{red}{y = -4x \ + \ 21} \)

Writing Linear Equations Quiz