Question

What is the rate change, the initial value and the equation of the line that goes through the points $\left( 1,5 \right)$ and $\left( 2,7 \right)$ ?

Answer 1

To solve the question we need to know the concept of line equation. First step is to find the slope which is the rate change. To find the slope the formula used is $Slope=\dfrac{\vartriangle y}{\vartriangle x}$ . The next step is to find the equation of the line in slope-intercept form which is $y=mx+c$. By putting the coordinates given. We will find the equation of the line. The last step will be to find the initial value which is the value of $c$.

Complete step by step answer:
The question asks us to find the rate of change of a line, initial value of the equation of line and the equation of the line for the points $\left( 1,5 \right)$ and $\left( 2,7 \right)$. Firstly we will find the rate change which means the slope of the line. Slope of a line is formulated as the ratio of the change in $y$coordinates to change in $x$ coordinates. Mathematically it would be written as:
$Slope=\dfrac{\vartriangle y}{\vartriangle x}$
$\Rightarrow Slope=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
The coordinates are $\left( 1,5 \right)$ and $\left( 2,7 \right)$, so ${{y}_{1}}=5,{{y}_{2}}=7,{{x}_{1}}=1$ and ${{x}_{2}}=2$. Putting the values in the above formula we get:
$\Rightarrow Slope=\dfrac{7-5}{2-1}$
$\Rightarrow Slope=\dfrac{2}{1}$
$\Rightarrow Slope=2$
So the rate of change is $2$.
Now the second part of the question is to find the initial value of the line but for this we need to find the value of the equation of the line, first. To find the equation of the line we will use slope- intercept form which is equal to $y=mx+c$ , where $''c''$ is the constant, $''m''$ is the slope of the line. To find the equation of the line we will find the intercept form:
$y=mx+c$
We have the value of slope which is $m$ as $2$. Since the points $\left( 1,5 \right)$ and $\left( 2,7 \right)$are in the line, we can put any of the coordinates to find the value of $c$. We will put the coordinate $\left( 1,5 \right)$ in the above equation:
$\Rightarrow y=2x+c$
$\Rightarrow 5=2\times 1+c$
Taking $2$to left hand side we get:
$\Rightarrow 5-2=c$
$\Rightarrow 3=c$
So the equation of the line thus formed is $y=2x+3$
The last part of the question is to find the initial value. The initial value is the constant of the line in the intercept form which is $c$ in this case.
So the initial value is $3$.
$\therefore$The rate change, the initial value and the equation of the line that goes through the points $\left( 1,5 \right)$ and $\left( 2,7 \right)$ are $2,3$ and $y=2x+3$ respectively.

Note: The equation of the line could be found in many ways. One of the ways to is slope-intercept form which has been used here in this question. The rate of change of a line refers to the slope of the line. The initial value is the constant $c$ in the slope- intercept form of equation.