Question

How do I create a linear equation with a given solution like $ (3,0) $ ?

Answer 1

Hint: Here in this question, we have to create linear equation using a given point, to create linear equation use the equation of the line, which intersects the y-axis at a point is given by: \[y = mx + b\], Where, b is the intercept, m is the slope of the line and y and x indicate the points on y-axis and x-axis respectively. On substituting we get the required solution.

Complete step by step solution:
Linear equations are equations of the first order. These equations are defined for lines in the coordinate system. An equation for a straight line is called a linear equation. The general representation of the straight-line equation is \[y = mx + b\] where, m is the slope of the line and b is the y-intercept.
Now, we have to create a linear equation with a given solution of point $ (3,0) $ . Which means the line must pass through the point $ (3,0) $ . In other words, we can say that we need to find a slope and y-intercept so that the equation \[y = mx + b\] will only work when $ (x,y) = (3,0) $ .
Now, we substitute these values in the equation \[y = mx + b\] .
 $ \Rightarrow y = mx + b $
 $ \Rightarrow 0 = m(3) + b $
Now, we will solve for the value of $ b $ i.e., y-intercept
 $ \Rightarrow \,\,b = - 3m $
On substituting, y-intercept $ b $
Then, the equation will be,
 $ \Rightarrow y = mx - 3m $
Take \[m\] as common
 $ \Rightarrow y = m(x - 3) $
Ending with an equation like this that links \[b\] and \[m\] means that there are an infinite number of slope-intercept pairs that produce a line that passes through $ (3,0) $ .
Hence, there are infinitely many such linear equations. Choose any \[b\] or \[m\] and plug it into \[b = - 3m\] to get the other. Your equation will be \[y = mx + b\] with these values of \[m\] and \[b\] .
So, the correct answer is “ y = m(x - 3)”.

Note: The general or standard representation of the straight-line equation is \[Ax + By = C\], it involves only a constant term and a first-order (linear) term. While shifting or transforming the term in the equation we should take care of the sign. Here sign conventions are used in this problem.