Question

Solve the following equations by using substitution method:
$10x+3y=75;6x-5y=11$

Answer 1

- Hint: It is given we must use a substitution method. First, look at the substitution method definition carefully and understand it. Try to convert the variable y in terms of x from any equation. Now substitute the value of y back into the remaining equation. By this you get an equation which has only one variable. It is called a single variable equation. Try to keep all variable terms on the left hand side and all constants on the right hand side. Algebraically, find the value of a variable. Using the value, as you know relation with other variables just substitute it to get the value of that variable. The pair of values will be your result. Just verify them by substituting into one of the equations.

Complete step-by-step solution -

Substitution Method: The method of solving a system of equations. It works by solving one of the equations for one of the variables to get in terms of another variable, then plugging this back into other equation, and solving for the other variable. By this you can find both the variables. The method is generally used when there are 2 variables. For more variables it will be tough to solve.
Given expression in the question which we need to solve are given by:
$10x+3y=75$ …………………..(i) $6x-5y=11$ ………………………………..(ii)
By dividing with 5y on both sides of equation (ii), we get
$\Rightarrow 6x=11+5y$
By dividing with 6 on both sides of above equation, we get:
$\Rightarrow \dfrac{6x}{6}=\dfrac{11+5y}{6}$
By simplifying the above equation, we get:
$\Rightarrow x=\dfrac{11}{6}+\dfrac{5y}{6}$ …………………………(iii)
By substituting equation (iii) in equation (i), we get the equation:
$\Rightarrow 10\left( \dfrac{11}{6}+\dfrac{5y}{6} \right)+3y=75$
We multiply constant 10 inside the bracket, to remove the bracket:
$\Rightarrow 10\cdot \dfrac{11}{6}+\dfrac{5y}{6}\cdot 10+3y=75$
By Taking least common multiple on left hand side, we get
$\Rightarrow \dfrac{110+50y+18y}{6}=75$
By multiplying with 6 on both sides of equation, we get:
$\Rightarrow 110+50y+18y=75\times 6$
By subtracting 110 on both sides of the equation, we get:
$\Rightarrow 50y+18y=75\times 6-110$
By simplifying the terms on both sides of above equation, we get:
$\Rightarrow 68y=340$
By dividing with 68 on both the sides, we get it as:
$\Rightarrow \dfrac{68y}{68}=\dfrac{340}{68}$
By simplifying the above equation, we get value of y as:
$\Rightarrow y=5$
By substituting this value into equation (iii), we get x as:
$\Rightarrow x=\dfrac{11}{6}+\dfrac{\left( 5 \right)\left( 5 \right)}{6}$
By simplifying the term in the above equation, we get:
$\Rightarrow x=\dfrac{11+25}{6}$
By simplifying the calculation of x, we get the value of x as:
$\Rightarrow x=6$
By substituting $x=6,y=5$ in equation (ii), we get:
$10\left( 6 \right)+3\left( 5 \right)=75$
By simplifying left hand side of above equation, we get:
$75=75$
 Hence, Verified.
Therefore, the solution of given equations as $\left( 6,5 \right)$ .

Note: Be careful while removing brackets. Don’t forget that the constant must also be multiplied. Generally, students multiply variables and forget about constant. Verification of a solution must be done to prove that our result is correct. Similarly, you can first find x in terms of y and then substitute and continue. Anyways, you will get the same result because the values of x, y won’t change.