Question

How do you solve the system using the elimination method for \[-3x+2y=-6\] and \[2x+2y=5\]?

Answer 1

Hint: Now we are given two linear equations. Now to solve the equations we will first multiply the equations such that the coefficient of one variable in both equations is the same. Now we will add or subtract the equations so that we eliminate the variable. Now we have a linear equation in one variable and hence we will solve the equation to find the value of the variable. Now substituting this in any equation we get the value of another variable.

Complete step by step answer:
For the given problem we are given to solve the system using the elimination method for equations \[-3x+2y=-6\] and \[2x+2y=5\].
Now let us consider the two equations and equation (1) and equation (2).
\[-3x+2y=-6...........\left( 1 \right)\]
Second equation will be
\[2x+2y=5............\left( 2 \right)\]
In the question it is given that we have to solve the problem using elimination methods. Which means you either add or subtract the equations to get an equation in one variable.
To get the equation in one variable we have to eliminate the other variable i.e. we have to make one of the variable’s coefficients should be in opposite in the two equations.
By observing the two equations we can say that the coefficient of y is same in both 2 equations.
Let us multiply it with ‘-sign’ to any of the equations.
Multiplying the equation () with ‘-sign’, we get
\[\Rightarrow 3x-2y=6\]
Let us consider the equation as equation (2), we get
\[\Rightarrow 3x-2y=6............\left( 3 \right)\]
Now let us add the both equations (2) and (3), we get
\[\Rightarrow 2x+2y+3x-2y=5+6\]
By simplifying the above equation, we get
\[\Rightarrow 5x=11\]
\[\Rightarrow x=\dfrac{11}{5}\]
Let us consider the above equation as equation (4) and substitute in equation (1), we get
\[\Rightarrow x=\dfrac{11}{5}.............\left( 4 \right)\]
\[\Rightarrow -3\left( \dfrac{11}{5} \right)+2y=-6\]
By simplifying the above equation, we get
\[\begin{align}
  & \Rightarrow 2y=\dfrac{33}{5}-6 \\
 & \Rightarrow y=\dfrac{3}{5\times 2} \\
 & \Rightarrow y=\dfrac{3}{10} \\
 & \\
\end{align}\]
Let us consider it as equation (5), we get
\[\Rightarrow y=\dfrac{3}{10}..........\left( 5 \right)\]
Therefore, equation (4) and (5) are solutions for the given problem.

Note: Now we can also solve the equation by substitution method. To solve the equation we will write one variable in terms of another using any one equation. Hence let us consider \[2x+2y=5\] . Now rearranging the terms we get, $x=\dfrac{5-2y}{2}$ . Now we will substitute the value in another equation and hence we get the value of y. Now again we substitute the value of y in any equation to find x. Hence we get the solution of the equation.