# Solving linear function

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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.

## Example:

Find an equation of the line through the given pair of points.

(-1, -3) and (-4, -7)

### Solution:

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 $y = \frac{4}{3}x - \frac{5}{3}$