Question

How do you solve \[y = \dfrac{2}{3}x - 7\] and \[2x - 3y = 21\] by graphing?

Answer 1

Hint: As there are two equations involved, in which we need to solve for x and y. To solve the given simultaneous equation, combine all the like terms or by using any of the elementary arithmetic functions i.e., addition, subtraction, multiplication and division hence simplify the terms to get the value of \[x\] also the value of \[y\] and plot the graph.

Complete step by step solution:
Let us write the given function:
 \[y = \dfrac{2}{3}x - 7\] …………………………. 1
 \[2x - 3y = 21\] ..……………………… 2
We need to plot the straight lines: \[y = \dfrac{2}{3}x - 7\] and \[2x - 3y = 21\] by solving the value of x and y.
The standard form of simultaneous equation is
 \[Ax + By = C\]
Hence, substitute the value of \[y\] in equation 2 as
 \[2x - 3y = 21\]
 \[ \Rightarrow 2x - 3\left( {\dfrac{2}{3}x - 7} \right) = 21\]
After substituting the y term, simplify the obtained equation by multiplying the terms as directed in the equation as:
 \[2x - \dfrac{6}{3}x + 21 = 21\]
Simplifying the terms again, we get:
 \[ \Rightarrow 2x - 2x + 21 = 21\]
As, we know that, \[2x - 2x = 0\] , hence we get:
 \[ \Rightarrow 21 = 21\]
Hence, the solution we got up is with a true equation but no variable solution. It means that the original equations were not independent; they are just rearranged versions of each other.
Graphically the two equations are just different representations of the same graph, and therefore, do not have one definite answer. In terms of a graph the two equations represent the same line and therefore do not intersect to give a single solution. Hence, there is no single solution to the system of equations.

Note: We know that Simultaneous equations are two equations, each with the same two unknowns and are "simultaneous" because they are solved together, hence the key point to solve these kinds of equations we need to combine all the terms and then simplify the terms to get the value of \[x\] also the value of \[y\] but here the given equations are identical and therefore, represent the same equation when we solve for x and y.