Question

How do you solve y=5x-7 and -3x-2y=-12 by substitution?

Answer 1

Hint: This type of question is based on the concept of linear equations with two variables. We should solve both the equations by substitution method. Consider the second equation and substitute the value of y=5x-7 which is the first equation. Now, we get an equation with one variable x. solve x by making necessary calculations. Then, substitute the value of x in the first equation y=5x-7 and find the value of y.

Complete step-by-step solution:
According to the question, we are asked to solve y=5x-7 and -3x-2y=-12 by substitution.
We have been given the equations are y=5x-7 ---------(1)
and -3x-2y=-12. -----------(2)
First consider equation (2).
-3x-2y=-12
By we know y=5x-7 from equation (1).
Let us substitute the value of y in equation (2) to obtain an equation with one variable x.
\[\Rightarrow -3x-2\left( 5x-7 \right)=-12\]
We have to use distributive property \[a\left( b+c \right)=ab+ac\] in the above obtained equation.
\[\Rightarrow -3x-2\times 5x-2\times -7=-12\]
On further simplifications, we get
\[\Rightarrow -3x-10x-2\times -7=-12\]
\[\Rightarrow -3x-10x+14=-12\]
Now group all the x terms from the left-hand side of the equation. We get
\[\left( -3-10 \right)x+14=-12\]
On further simplifications, we get
\[\Rightarrow \left( -13 \right)x+14=-12\]
Now, subtract 14 from both the sides of the equation.
\[\Rightarrow \left( -13 \right)x+14-14=-12-14\]
Since terms with same magnitude and opposite signs cancel out, we get
\[\left( -13 \right)x=-12-14\]
\[\Rightarrow \left( -13 \right)x=-26\]
Now divide the whole equation by -13. We get
\[\dfrac{-13x}{-13}=\dfrac{-26}{-13}\]
On further simplifications, we get
\[\Rightarrow \dfrac{-13x}{-13}=\dfrac{-13\times 2}{-13}\]
We find that -13 is common in both the numerator and denominator in both the sides of the equation. Let us now cancel -13 from the equation.
\[\Rightarrow x=\dfrac{2}{1}\]
\[\therefore x=2\]
Now we have to find the value of y.
Substitute the value of x in equation (1), we get
$y=5(2)-7$
On further simplifications, we get
$y=10-7$
Therefore, y=3.
Hence, the value of x and y in the given system y=5x-7 and -3x-2y=-12 are 2 and 3 respectively.

Note: We can check whether the obtained values are correct or not.
First, consider the equation (3), that is y=5x-7.
LHS=$-3x-2y$
LHS=-3(2)-2(3) [since x=2 and y=3]
On further simplifications, we get
LHS=$-6-6$
LHS=$-12$
Now consider RHS.
RHS=$-12$
Therefore, LHS=RHS.
It is enough to check with one equation.
Similarly, we can check for the other equation also.
Hence, the obtained answer is correct.