Question

How do you solve this set of linear systems: y=9x+17; y=x+49?

Answer 1

Hint: This type of problem is based on the concept of linear equations with two variables. Here, we find that both the linear equations in the system have the same LHS, that is y. equate the LHS of both the equations. Then, subtract x and 17 from both the sides of the equation. We get an equation with variable x in the LHS. Divide the whole equation by 8 to find the value of x. then, substitute the value of x in y=x+49 and obtain the value of y.

Complete step-by-step solution:
According to the question, we are asked to solve the set of linear systems: y=9x+17; y=x+49.
 We have been given the two equation are
y=9x+17 ---------(1) and
y=x+49. ---------(2)
From equation (1) and (2), we find the LHS is the same that is y.
Since, the left-hand sides of the equations are the same, the right-hand sides should also be the same.
Therefore, equating the right-hand sides of the equation, we get
\[\Rightarrow 9x+17=x+49~~\] --------(3)
Subtract x from both the sides of the equation, we get
\[9x+17-x=x+49~-x~\]
We know that the terms with the same magnitude and opposite signs cancel out.
On cancelling x from the RHS, we get
\[\Rightarrow 9x+17-x=49~~\]
Now group all the x terms in the LHS.
\[\Rightarrow x\left( 9-1 \right)+17=49\]
On further simplifications, we get
8x+17=49
Now subtract 17 from both the sides of the equation. We get
\[\Rightarrow 8x+17-17=49-17\]
We know that the terms with the same magnitude and opposite signs cancel out.
On cancelling 17 from the LHS, we get
8x=49-17
On further simplification, we get
8x=32
Now, we need to find the value of x.
Divide the whole equation by 8.
\[\Rightarrow \dfrac{8x}{8}=\dfrac{32}{8}\]
We find that 8 are common in the numerator and denominator of LHS. On cancelling 8, we get
\[x=\dfrac{32}{8}\]
We can write 32 as the product of 4 and 8.
\[\Rightarrow x=\dfrac{4\times 8}{8}\]
We find that 8 are common in the numerator and denominator of RHS. On cancelling 8, we get
\[x=4\]
Now, we need to find the value of y.
Substitute the value of x in equation (2).
\[\Rightarrow y=4+49\]
On further simplification, we get
y=53.
Hence, the value of x and y in the given set of equations y=9x+17 and y=x+49 are 4 and 53 respectively.

Note: We can verify whether the answer obtained is correct or not.
Substitute the values of x and y in the given system and check whether the LHS is equal to RHS.
Consider equation (1), that is y=9x+17.
Here, LHS=y
But we know that y=53.
Therefore, LHS=53.
Now consider the RHS.
RHS=9x+17
We know that x=4,
Therefore, RHS=9(4)+17
On further simplifications, we get
RHS=36+17
RHS=53
Here, LHS=RHS.
It is enough to check in one linear equation.
Hence the obtained answer is correct.