If $4x + \dfrac{6}{y}= 15 $ and $6x - \dfrac{8}{y} = 14$ , then the value of p in y = px - 2 is :

Question

If $4x + \dfrac{6}{y}= 15 $ and  $6x - \dfrac{8}{y} = 14$ , then the value of p in y = px - 2 is :

Prepp Team · Accepted Answer

Solving System of Equations to Find Parameter Value

The problem asks us to first solve a given system of two equations for the variables $x$ and $y$. Once we find the values of $x$ and $y$, we need to substitute them into a third equation, $y = px - 2$, and find the value of the parameter $p$.

Given System of Equations

The system of equations is:

$4x + \dfrac{6}{y}= 15$
$6x - \dfrac{8}{y} = 14$

Solving the System Using Elimination Method

We can solve this system of equations using the elimination method. Notice that the terms involving $y$ are fractions. Let's eliminate the terms with $\dfrac{1}{y}$.

To eliminate $\dfrac{1}{y}$, we can multiply the first equation by a suitable number and the second equation by another suitable number so that the coefficients of $\dfrac{1}{y}$ become equal in magnitude but opposite in sign. The least common multiple of 6 and 8 is 24.

Multiply equation (1) by 4:

$4 \times (4x + \dfrac{6}{y}) = 4 \times 15$

$16x + \dfrac{24}{y} = 60$ (Equation 3)

Multiply equation (2) by 3:

$3 \times (6x - \dfrac{8}{y}) = 3 \times 14$

$18x - \dfrac{24}{y} = 42$ (Equation 4)

Now, we can add Equation 3 and Equation 4 to eliminate the $\dfrac{1}{y}$ term:

$(16x + \dfrac{24}{y}) + (18x - \dfrac{24}{y}) = 60 + 42$

$16x + 18x + \dfrac{24}{y} - \dfrac{24}{y} = 102$

$34x = 102$

Now, solve for $x$:

$x = \dfrac{102}{34}$

$x = 3$

Finding the Value of y

Now that we have the value of $x$, we can substitute $x=3$ into either of the original equations to find the value of $y$. Let's use equation (1):

$4x + \dfrac{6}{y}= 15$

Substitute $x=3$:

$4(3) + \dfrac{6}{y}= 15$

$12 + \dfrac{6}{y}= 15$

Subtract 12 from both sides:

$\dfrac{6}{y}= 15 - 12$

$\dfrac{6}{y}= 3$

To solve for $y$, we can multiply both sides by $y$ and then divide by 3, or simply see that if $\dfrac{6}{y}= 3$, then $y$ must be $\dfrac{6}{3}$:

$6 = 3y$

$y = \dfrac{6}{3}$

$y = 2$

So, the solution to the system of equations is $x=3$ and $y=2$.

Finding the Value of p

We are given the equation $y = px - 2$. We have found that $x=3$ and $y=2$. We need to substitute these values into this equation and solve for $p$.

$y = px - 2$

Substitute $y=2$ and $x=3$:

$2 = p(3) - 2$

$2 = 3p - 2$

Add 2 to both sides of the equation:

$2 + 2 = 3p$

$4 = 3p$

Divide both sides by 3 to solve for $p$:

$p = \dfrac{4}{3}$

The value of $p$ is $\dfrac{4}{3}$.

Revision Table: Key Steps

Step	Action	Result
1	Solve the system of equations for $x$ and $y$.	$x=3$, $y=2$
2	Substitute $x$ and $y$ values into $y = px - 2$.	$2 = p(3) - 2$
3	Solve the resulting equation for $p$.	$p = \dfrac{4}{3}$

Additional Information: Solving Systems

A system of equations is a set of two or more equations that share variables. The solution to a system is the set of values for the variables that makes all equations in the system true simultaneously.

Common methods to solve a system of linear equations include:

Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
Elimination Method: Multiply one or both equations by constants so that the coefficients of one variable are opposite. Then add the equations to eliminate that variable. This is the method used in the solution above.
Graphical Method: Graph each equation on the same coordinate plane. The intersection point(s) represent the solution(s). This method is less precise for non-integer solutions.

The given system involved terms with $\dfrac{1}{y}$. While not strictly linear, treating $x$ and $\dfrac{1}{y}$ as variables allows us to use linear methods like elimination or substitution effectively.

Prepp Team · Answer

4

Prepp Team · Answer

3/4

Prepp Team · Answer

3

Step	Action	Result
1	Solve the system of equations for \(x\) and \(y\).	\(x=3\), \(y=2\)
2	Substitute \(x\) and \(y\) values into \(y = px - 2\).	\(2 = p(3) - 2\)
3	Solve the resulting equation for \(p\).	\(p = \dfrac{4}{3}\)