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Question: How do you solve \(\dfrac{3}{4}x+\dfrac{1}{3}y=1\) and \(x-y=10\)?...

How do you solve 34x+13y=1\dfrac{3}{4}x+\dfrac{1}{3}y=1 and x−y=10x-y=10?

Explanation

Solution

There are two unknowns xx and yy and also two equations to solve. We solve the equations equating the coefficients of one variable and omitting the variable. The other variable remains with the constants. Using the binary operation, we find the value of the other variable. First, we are applying the process of reduction and then the substitution.

Complete step by step solution:
Simplifying the equation 34x+13y=1\dfrac{3}{4}x+\dfrac{1}{3}y=1, we get 9x+4y=129x+4y=12
The given equations 9x+4y=129x+4y=12 and x−y=10x-y=10 are linear equations of two variables.
We know that the number of equations has to be equal to the number of unknowns to solve them.
We take the equations as 9x+4y=12.....(i)9x+4y=12.....(i) and x−y=10......(ii)x-y=10......(ii).
We multiply 4 to the both sides of the second equation and get
4×(x−y)=4×10 ⇒4x−4y=40 \begin{aligned} & 4\times \left( x-y \right)=4\times 10 \\\ & \Rightarrow 4x-4y=40 \\\ \end{aligned}
We take the equation as 4x−4y=40.....(iii)4x-4y=40.....(iii).
Now we add the equation (i) to equation (iii) and get
(9x+4y)+(4x−4y)=12+40\left( 9x+4y \right)+\left( 4x-4y \right)=12+40.
We take the variables together and the constants on the other side.
Simplifying the equation, we get
(9x+4y)+(4x−4y)=12+40 ⇒13x=52 ⇒x=4 \begin{aligned} & \left( 9x+4y \right)+\left( 4x-4y \right)=12+40 \\\ & \Rightarrow 13x=52 \\\ & \Rightarrow x=4 \\\ \end{aligned}
The value of xx is 4. Now putting the value in the equation x−y=10......(ii)x-y=10......(ii), we get
x−y=10 ⇒y=4−10=−6 \begin{aligned} & x-y=10 \\\ & \Rightarrow y=4-10=-6 \\\ \end{aligned}.
Therefore, the values are x=4,y=−6x=4,y=-6.

Note: We can also find the value of one variable yy with respect to xx based on the equation
x−y=10x-y=10 where y=x−10y=x-10. We replace the value of yy in the second equation of
9x+4y=129x+4y=12 and get

& 9x+4y=12 \\\ & \Rightarrow 9x+4\left( x-10 \right)=12 \\\ & \Rightarrow 9x+4x-40=12 \\\ \end{aligned}$$ We get the equation of $x$ and solve $\begin{aligned} & 9x+4x-40=12 \\\ & \Rightarrow 13x=52 \\\ & \Rightarrow x=4 \\\ \end{aligned}$ Putting the value of $x$ we get $x-y=10\Rightarrow y=4-10=-6$. Therefore, the values are $x=4,y=-6$.