Question

How do you solve this system of equations: $12x-5y=30$ and $y=2x-6$ ?

Answer 1

Hint: We are given two equations which must be solved simultaneously as well as plotted on the same graph to find the solution of these equations. In order to plot any equation on a graph, we must find at least two points lying on it which can be further marked and joined to sketch the graph. Hence, we shall first find the points on these functions and then find their solutions.

Complete step by step answer:
We will put the values of x and y equal to zero one by one to find two simple points one of which will have its x-coordinate equal to zero and the other one would have its y-coordinate equal to zero.
We shall first find the points lying on the line whose equation is given by, $12x-5y=30$.
Putting $x=0$in the equation, we get
$12\left( 0 \right)-5y=30$
$\Rightarrow -5y=30$
Now, we shall take -5 to right-hand side and divide to find y:
$\Rightarrow y=\dfrac{30}{-5}$
$\therefore y=-6$
Therefore, we get the point as $\left( 0,-6 \right)$.
Putting $y=0$in the equation, we get
$12x-5\left( 0 \right)=30$
$\Rightarrow 12x=30$
Now, we shall take 12 to right-hand side and divide to find x:
$\Rightarrow x=\dfrac{30}{12}=\dfrac{5}{2}$
$\therefore x=\dfrac{5}{2}$
Therefore, we get the point as $\left( \dfrac{5}{2},0 \right)$.
Hence, the points are $\left( 0,-6 \right)$and $\left( \dfrac{5}{2},0 \right)$. ……………………….. (1)
We shall now find the points on second equation given as, $y=2x-6$.
Putting $x=0$in the equation, we get
$y=2\left( 0 \right)-6$
$\Rightarrow y=-6$
Therefore, we get the point as $\left( 0,-6 \right)$.
Putting $y=0$in the equation, we get
$\left( 0 \right)=2x-6$
$\Rightarrow 2x=6$
$\therefore x=3$
Therefore, we get the point as $\left( 3,0 \right)$.
Hence, the points are $\left( 0,-6 \right)$and $\left( 3,0 \right)$. ……………………… (2)
From (1) and (2), we get the graph as:

So the common point is the solution of pairs of equations.

Note: While plotting the graph, the points must be carefully marked. We often tend to get confused between the two coordinates and make the points incorrectly. Also, while sketching any graph, the two points should always be taken such that the x or y coordinate is zero in them because it makes the calculations easier.