Solve the system of equations graphically.
x+y=3, 2x+5y=12.
VerifiedHint: Be clear that the two equations given in the question are linear equations in two variables, x and y. Substituting values of x in the first equation, you will be able to get corresponding values of y. When you have two such sets of points, then you can plot the line. The same procedure has to be done for the second equation also. While you will represent them geometrically you will get two different lines on the Cartesian plane. The solution to the system of equations is the meet point of the two lines on the Cartesian plane.
Complete step-by-step answer:
Before starting with the solution, let us discuss the significance of linear equations in two variables. Linear equations in two variables represent the equation of a line on the Cartesian plane.
Now we will move to the solution to the given question. First, let us see the equation x + y = 3.
Now, as this is a linear equation, it represents a line. In order to represent this in a Cartesian plane, we convert the equation of line to the Cartesian form. On doing so, we get
x + y = 3
\[\Rightarrow \dfrac{x}{3}\text{ }+\text{ }\dfrac{y}{3}\text{ }=\text{ }1\]
Therefore, the x-intercept of the line is 3, and the y-intercept is 3. So, the line will meet the x-axis and the y-axis at (3,0) and (0.3).
Similarly, let us move to the equation 2x + 5y = 12.
Now, as this is a linear equation, it represents a line. In order to represent this in a Cartesian plane, we convert the equation of line to the Cartesian form. On doing so, we get
2x + 5y = 12
\[\Rightarrow \dfrac{x}{6}\text{ }+\text{ }\dfrac{y}{\dfrac{12}{5}}\text{ }=\text{ }1\] .
Therefore, the x-intercept of the line is 6, and the y-intercept is $\dfrac{12}{5}$ . So, the line will meet the x-axis and the y-axis at (6,0) and $\left( 0,\dfrac{12}{5} \right)$ .
Now we will represent the above lines on a cartesian plane. On doing so, we get
The meet point of the lines, i.e., the solution of the system of equations is (1,2), i.e., x=1 and y=2.
Note: Be careful with the signs and calculations as in such questions, the possibility of making a mistake is either of the sign or a calculation error. Also, we should be clear that a solution can represent different geometries when represented in planes with different dimensions, and all the points lying on these geometries would represent a solution to the linear equation. For example: when the solution of the above equation is represented on a number line, i.e., a single dimensioned plane, the geometry formed is just a point while on the Cartesian plane, it is represented by a straight line. Further, if we extend it to a 3-D plane, we will find the same solution will represent a plane.
