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# How do you solve $3x+2y=5$ and $2x+5y=-3$?

Last updated date: 13th Jun 2024
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Hint: We can solve this question using basic linear equation concepts. We can make them a layered problem. We align the variables under each other, and then use algebra to manipulate the expressions to eliminate one unknown and solve for the other, Then we take that solved variable and use it to find the second one. Then we can find values of both the variables.

Complete step by step solution:
Given equations are
$3x+2y=5....\left( 1 \right)$
$2x+5y=-3....\left( 2 \right)$
Here we can see that we don’t have equal and opposite coefficients so there is a need to make changes on equations.
If we don’t have any terms with equal coefficients then we have to make any one term to be equal by multiplying or dividing. If both of them are not having opposite signs we have to subtract otherwise we have to add.
Here we will multiply our equation 1 with 2 and equation 2 with 3 to make equal coefficients. By doing this we will get
$6x+4y=10$
$6x+15y=-9$
Here we have the same signs for both the coefficients so we will subtract the equations.
By subtracting both the equations we get
\begin{align} & \underline{\begin{align} & 6x+4y=10 \\ & 6x+15y=-9 \\ \end{align}} \\ & -11y=19 \\ \end{align}
Now we have to divide the equation with 11 on both sides.
$\Rightarrow -\dfrac{11y}{11}=\dfrac{19}{11}$
We will get
$\Rightarrow -y=\dfrac{19}{11}$
Now we will multiply with – on both sides we will get
$y=-\dfrac{19}{11}$
Now we will substitute this y value in the first equation given to get the x value.
Our equation 1 is
$3x+2y=5$
Now we have to substitute y value in it
We will get
$\Rightarrow 3x+2\left( -\dfrac{19}{11} \right)=5$
$\Rightarrow 3x-\dfrac{38}{11}=5$
Now grouping like terms on the side we can solve the equation.
To do this we have to add $\dfrac{38}{11}$ to both sides of the equation.
$\Rightarrow 3x-\dfrac{38}{11}+\dfrac{38}{11}=5+\dfrac{38}{11}$
$\Rightarrow 3x=5+\dfrac{38}{11}$
By simplifying we get
$\Rightarrow 3x=\dfrac{55+38}{11}$
$\Rightarrow 3x=\dfrac{93}{11}$
Now we have to divide the equation with 3 on both sides.
$\Rightarrow \dfrac{3x}{3}=\dfrac{\dfrac{93}{11}}{3}$
By simplifying we will get
$\Rightarrow \dfrac{3x}{3}=\dfrac{93}{33}$
$\Rightarrow x=\dfrac{31}{11}$
So the x and y values by solving the equations given is
$x=\dfrac{31}{11},y=-\dfrac{19}{11}$

Note: We can also check the answer by back substituting the values in the equation. We can substitute values in equation 2 and check whether it satisfies it or not. We can also do this by making y coefficients equal.