Question

How do you write an equation given $\left( { - 2,4} \right)$; $m = 3$?

Answer 1

In this question, we have to find the equation of a line which has slope $3$ and passes through the point $\left( { - 2,4} \right)$. It can be done by first finding the value of $c$ using the formula for the equation of a line. For this, substitute the value of $m,x,y$ into the equation. Next, move all terms not containing $c$ to the right side of the equation. Now that the values of $m$ (slope) and $c$ (y-intercept) are known, substitute them into $y = mx + c$ to find the equation of the line. The equation obtained will be the required equation of the given line in slope intercept form.

Formula used:
The Slope Intercept Form of a Line:
The equation of a line with slope $m$ and making an intercept $c$ on $y$-axis is $y = mx + c$.

Complete step by step solution:
We know that the slope intercept form of a line is the equation of a line with slope $m$ and making an intercept $c$ on $y$-axis is $y = mx + c$.
Given: $\left( { - 2,4} \right)$; $m = 3$
So, we have to write an equation in the form of $y = mx + c$, the equation of a line with slope $m$ and making an intercept $c$ on $y$-axis.
First, find the value of $c$ using the formula for the equation of a line.
Now, use the formula for the equation of a line to find $c$.
$y = mx + c$
Substitute the value of $m$ into the equation, we get
$\Rightarrow y = 3x + c$
Substitute the value of $x$ into the equation, we get
$\Rightarrow y = 3\left( { - 2} \right) + c$
Substitute the value of $y$ into the equation, we get
$\Rightarrow 4 = 3\left( { - 2} \right) + c$
Next, find the value of $c$.
For this, rewrite the equation as $3\left( { - 2} \right) + c = 4$.
$\Rightarrow 3\left( { - 2} \right) + c = 4$
Multiply $3$ by $- 2$, we get
$\Rightarrow - 6 + c = 4$
Now, move all terms not containing $c$ to the right side of the equation.
Add $6$ to both sides of the equation.
$\Rightarrow c = 4 + 6$
Add $4$ and $6$.
$\therefore c = 10$
Now that the values of $m$ (slope) and $c$ (y-intercept) are known, substitute them into $y = mx + c$ to find the equation of the line.
$\Rightarrow y = 3x + 10$

Final solution: Hence, the equation of the line is $y = 3x + 10$.

Note: We can directly find the equation of line by substituting ${x_1} = - 2$, ${y_1} = 4$ and $m = 3$ in equation of the line $y - {y_1} = m\left( {x - {x_1}} \right)$.
$\Rightarrow y - 4 = 3\left( {x + 2} \right)$
Now, apply distributive property in the above equation, i.e., multiplying each addend individually by the number and then adding the products together.
$\Rightarrow y - 4 = 3x + 6$
Now we have to isolate the variable term, $y$ on one side by performing the same mathematical operations on both sides of the equation.
So, adding $4$ to both sides of the equation, we get
$\Rightarrow y - 4 + 4 = 3x + 6 + 4$
It can be written as
$\Rightarrow y = 3x + 10$
Final solution: Hence, the equation of the line is $y = 3x + 10$.