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Question: How do you solve the following system \(x+2y=9\), \(7x+15y=32\)?...

How do you solve the following system x+2y=9x+2y=9, 7x+15y=327x+15y=32?

Explanation

Solution

Now the given equations are linear equations in two variables. To solve the equations we will multiply each equation with an appropriate number such that the coefficient of x in both equations is the same. Now we will subtract the two equations and find the value of y. Now substituting the value of y in any equation we can find the value if x.

Complete step by step solution:
Now we are given with two linear equations in two variables x and y.
Now we know that linear equations in two variables are nothing but straight lines in the XY plane.
Now we want to find the solution of the equation. We know that the solution of the linear equation is the intersection point of the two lines. Now let us solve the linear equation simultaneously.
Now consider the equation x+2y=9x+2y=9.
Now multiplying the equation by 7 so that the coefficient of x for both equations are the same. Hence we get,
⇒7x+14y=63\Rightarrow 7x+14y=63
Now subtracting the equation obtained from the equation 7x+15y=327x+15y=32 .
Hence we get the equation as
⇒7x+14−7x−15y=63−32 ⇒14−15y=32 \begin{aligned} & \Rightarrow 7x+14-7x-15y=63-32 \\\ & \Rightarrow 14-15y=32 \\\ \end{aligned}
Now rearranging the term of the equation we get,
⇒14−32=15 ⇒−18=15y ⇒y=−1815 \begin{aligned} & \Rightarrow 14-32=15 \\\ & \Rightarrow -18=15y \\\ & \Rightarrow y=\dfrac{-18}{15} \\\ \end{aligned}
Hence the value of y is −1815\dfrac{-18}{15} . Now substituting the value of y in the equation x+2y=9x+2y=9
⇒x+2×−1815=9 ⇒x+−3615=9 ⇒x=9+3615 ⇒x=15×9+3615 \begin{aligned} & \Rightarrow x+2\times \dfrac{-18}{15}=9 \\\ & \Rightarrow x+\dfrac{-36}{15}=9 \\\ & \Rightarrow x=9+\dfrac{36}{15} \\\ & \Rightarrow x=\dfrac{15\times 9+36}{15} \\\ \end{aligned}
⇒x=17115 ⇒x=575 \begin{aligned} & \Rightarrow x=\dfrac{171}{15} \\\ & \Rightarrow x=\dfrac{57}{5} \\\ \end{aligned}
Hence the solution of the equation is x=575x=\dfrac{57}{5} and y=−1815y=\dfrac{-18}{15}.

Note: Now note that we can also solve the equation by method of substitution. To solve the equation we will consider any equation and try to write y in terms of x. then we substitute the value of y in the equation and hence we get a linear equation in x. Hence using this we find the value of x. Further on substituting the value in any equation we can find the value of y.