Question

The line of the equation \[y = mx + c\] passes through the point (1,4) and (2,5) determine the value of m and c.

Answer 1

Hint: The definition of a linear equation is an algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line. An example of linear equation is \[y = mx{\text{ }} + {\text{ }}b.\]
In mathematics, the linear equation is an equation that may be put in the form of
\[{a_1}{x_1} + \cdots + {a_n}{x_n} + b = 0,\].
Where \[{x_1}, \ldots ,{x_n}{x_1}, \ldots ,{x_n}\] are the variables and \[b,{a_1}, \ldots ,{a_n}b,{a_1}, \ldots ,{a_n}\]are the variables, The coefficient may be considered as the parameters of the equation, and maybe arbitrary expression, provided they do not contains any variable.
One form of the equation of a straight line is called the slope intercept form because it contains information about these two properties.
The equation of a straight line
\[y = mx + c\]
Where m is slope of line (gradient)
C is vertical intercept (It is the value of y when \[x = 0\])

Complete step-by-step solution:
The line of equation \[y = mx + c...........(i)\]
Given point are \[(1,4)\] and \[(2,5)\]
Let points \[(1,4) = ({x_1},{y_1})\]
Points \[(2,5) = ({x_2},{y_2})\]
On substituting the values \[({x_1},{y_1})\]in \[e{q^n}\](i) where \[{x_1} = 1,\,\,\,{y_1} = 4\] we get
\[4 = m1 + c\]
\[4 = m + c......(ii)\]
On substituting the values \[{x_2},\,{y_2}\] in \[e{q^n}\](i) where \[{x_2} = 2,\,\,\,\,{y_2} = 5\] we get
\[5 = 2m + c……(iii)\]

Now solving \[e{q^n}\](ii) & \[e{q^n}\](iii) ( by elimination method)
\[\begin{gathered}
  \begin{array}{*{20}{c}}
  {\,4\,\, = \,\,m\,\, + \,\,c} \\
  {\underline {\,{}_ - 5\,\, = \,\,{}_ - 2m\,\, + \,\,c} }
\end{array} \\
  \,\,\, - 1\,\, = \,\, - \,m \\
\end{gathered} \]
$\Rightarrow$ \[m = 1\]

 By substituting value of\[\,m = 1\] in \[e{q^n}\](ii) and on solving we get,
\[ \Rightarrow 4 = 1 + c\]
\[\Rightarrow c = 4 - 1\] {transposing 1 on LHS}
\[c = 3\]

Note: Few points to remember regarding the above concepts are :
‘m’ can be positive, negative or zero.
Line with a positive slope upwards, from left to right
Line with \[( - m)\] slope downwards from left to right
Line with a zero gradient are horizontal
So, required \[m = 1\,\,\& \,c = 3\]
Required \[e{q^n}\] of line
\[y = mx + x\]
\[ \Rightarrow y = m + 3\].