The equation of the line is an algebraic representation of a set of points in a coordinate system that form a line.
It is usually written this way: y = mx + b.
y = how far up
x = how far along
m = Slope or Gradient (how steep the line is)
b = value of y when x=0
$m = \frac{Change in y}{Change in x}$
Now, we can find out the equation for a straight line using this information.
For example:
$m = \frac{2}{1} = 2$
b = 1 (value of y when x=0)
Putting it to y = mx + b gets us: y = 2x + 1.
Then, we can choose any value for x and find the matching value for y.
For instance, when x = 7,
y = 2×7 + 1 = 15
So when x = 7 you will have y = 15.
Find an equation of the line through the given pair of points.
(-1, -3) and (-4, -7)
Step1:
Write the equation of a linear function. Substitute
$\begin{cases}-3 = m \times (-1) + b\\-7 = m \times (-4) + b\end{cases}$
Step2:
Multiply the monomials
$\begin{cases}-3 = -m + b\\-7 = 4m + b\end{cases}$
Step3:
Rearrange unknown terms to the left side of the equation
$\begin{cases} m - b = 3\\4m - b = 7\end{cases}$
Step4:
Reorder the equation
$\begin{cases} -b + m = 3\\4m - b = 7\end{cases}$
$\begin{cases} -b + m = 3\\-b + 4m = 7\end{cases}$
Step5:
Subtract the equations
$-b + m - (-b + 4m) = 3 - 7$
Step6:
Remove parentheses
$-b + m + b - 4m = 3 - 7$
Cancel one variable
$m - 4m = 3 - 7$
Combine like terms
$-3m = 3 - 7$
Calculate the sum or difference
$-3m = -4$
Step7:
Divide both sides of the equation by the coefficient of the variable
$m = -4 \div (-3)$
Remove the parentheses
$m = -(-4 \div 3)$
Step8:
Rewrite as a fraction
$m = \frac{4}{3}$
Step9:
Substitute into one of the equations
$-b + \frac{4}{3} = 3$
Step10:
Solve equation
$b = -\frac{5}{3}$
The solution of the system is
$\begin{cases} b = -\frac{5}{3} \\ m=\frac{4}{3} \end{cases}$
Substitute
$y = \frac{4}{3}x - \frac{5}{3}$
The equation of the line is $y = \frac{4}{3}x - \frac{5}{3}$