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Question: How‌ ‌do‌ ‌you‌ ‌solve‌ ‌the‌ ‌system‌ ‌\(2x+5y=38\)‌ ‌and‌ ‌\(x-3y=-3\)‌ ‌using‌ ‌substitution?‌...

How‌ ‌do‌ ‌you‌ ‌solve‌ ‌the‌ ‌system‌ ‌2x+5y=382x+5y=38‌ ‌and‌ ‌x−3y=−3x-3y=-3‌ ‌using‌ ‌substitution?‌

Explanation

Solution

We first rewrite the second equation as x=−3+3yx=-3+3y . Then, we put this value of xx in the first equation and solve for yy . Then, having solved for yy , we put this value of yy in any one of the equations to get the value of xx .

Complete step-by-step solution:
The given equations that we have are
2x+5y=38....(1)2x+5y=38....\left( 1 \right)
x−3y=−3....(2)x-3y=-3....\left( 2 \right)
By substitution, we mean to rearrange one of the equations as a function of one variable like y=f(x)y=f\left( x \right) or x=f(y)x=f\left( y \right) . We then insert this rearranged equation in the other equation to get an equation having only a single variable. This equation can be solved to find the value of the variable. We then put this value of the variable in any one of the equations to get the value of the other variable.
Let us start by first rearranging the equation (2)\left( 2 \right) . We transfer the 3y3y term from the left hand side to the right hand side which gives,
⇒x=−3+3y....(3)\Rightarrow x=-3+3y....\left( 3 \right)
The above equation has been rearranged in the form x=f(y)x=f\left( y \right) . We now insert this function in equation (1)\left( 1 \right) and get,
2x+5y=382x+5y=38
⇒2(−3+3y)+5y=38 \Rightarrow 2\left( -3+3y \right)+5y=38\,
Using distributive property, we multiply the 22 with the terms inside the bracket and get,
⇒−6+6y+5y=38 \Rightarrow -6+6y+5y=38\,
Adding the yy terms, we get,
⇒−6+11y=38 \Rightarrow -6+11y=38\,
We add 66 to both sides of the above equation and get,
⇒11y=38 +6\Rightarrow 11y=38\,+6
Adding the arithmetic terms, we get,
⇒11y=44\Rightarrow 11y=44
Dividing both sides of the above equation by 1111 , we get,
⇒11y11=4411\Rightarrow \dfrac{11y}{11}=\dfrac{44}{11}
Simplifying the above equation, we get,
⇒y=4\Rightarrow y=4
We now put this value of yy in the equation (3)\left( 3 \right) to get,
⇒x=−3+3(4)\Rightarrow x=-3+3\left( 4 \right)
⇒x=9\Rightarrow x=9
Therefore, we can conclude that the solution of the given system of equations is x=9,y=4x=9,y=4 .

Note: We should be careful while rearranging the equation as a function of a single variable, by taking care of the signs. Also, when we substitute the function, the expression gets very large and thus most susceptible to mistakes. In the end, we should cross check our answer by putting the values in both the given equations.