Question

Solve $2x+3y=11$ and $2x-4y=-24$ and hence find the value of m for which $y=mx+3$ .

Answer 1

Hint: There are 2 variables, 2 equations. So it is best to use a substitution method. By using the definition of substitution method, try to convert the variable y in terms of x from any equation. By this you get an equation which has only x as variable. Try to keep all x terms on the left hand side and all constants on the right hand side. By this we get the value of variable x. By using the value, find y. Now substitute values of x, y in the last equation and try to solve for value of m. This m value is the required result.

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 another equation, and solving for the other variable. By this you can find both the variables. Then method is generally used when there are 2 variables. For more variables it will be tough to solve.
Given equations which we need to solve are given by:
$2x+3y=11$ …………………..(1)
$2x-4y=-24$ ……………………(2)
By dividing with 2 on both sides of equation (2), we get:
$x-2y=-12$
By adding 2y on both sides of above equations, we get:
$x-2y+2y=-12+2y$
By cancelling common terms of left hand side, we get it as:
$x=2y-12$ ……………………………………(3)
By substituting equation (3) into the equation (1), we get:
$\Rightarrow 2\left( 2y-12 \right)+3y=11$
By multiply constant into the bracket to remove it, we get:
$\Rightarrow 4y-24+3y=11$
By adding 24 on both sides of equation, we get:
$\Rightarrow 4x+3y=35$
By taking y common on left hand side of equation, we get:
$\Rightarrow \left( 4+3 \right)y=35$
By dividing with 7 on both sides, we get it as:
$\Rightarrow y=5$
By substituting this value into equation (3), we get x as:
$\Rightarrow x=2\left( 5 \right)-12=-2$
For verification, we substitute these values into the equation (2)
By substituting $x=-2,y=5$ into equation (2), we get:
$2\left( -2 \right)-4\left( 5 \right)=-24$
By simplifying the above equation, we get it as:
$\Rightarrow -24=-24$
So, LHS $=$ RHS. Hence, verified.
The solution of given equations is $\left( -2,5 \right)$ .
Next, we have other equation in terms of m as:
$\Rightarrow y=mx+3$
By substituting $\left( -2,5 \right)$ in this equation, we get it as:
$\Rightarrow 5=m\left( -2 \right)+3$
By subtracting 3 on both sides of equation, we get:
$\Rightarrow 5-3=m\left( -2 \right)$
By dividing with $-2$ on both sides, we get:
$\Rightarrow \dfrac{2}{-2}=m\dfrac{\left( -2 \right)}{\left( -2 \right)}$
By simplifying the above equation, we get it as:
$-1=m$
Therefore, the value of m to satisfy given conditions is $-1$ .

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.