Question

How do you solve the following system $5x - 7y = 22,4x - 3y = 15?$

Answer 1

Hint: Let us solve the system of equations by using elimination method.
In this system of equations multiplication is necessary to eliminate a variable.
After multiplication, add the two equations and solve for $y$.
Next substitute the value of $y$ in any one of the given equation to get the value of $x$

Complete step-by-step solution:
First mark the given two equations as $\left( 1 \right)$ and$\left( 2 \right)$.
The given equations are $5x - 7y = 22....\left( 1 \right)$ and $4x - 3y = 15\,...\left( 2 \right)$
Solve one of the equations for one of the variables. Let’s solve the first equation for y.
Now we have to eliminate $x$ in the system of equations.
Multiply the first equation by $4,$ and the second equation by $5$we get,
$\left( 1 \right) \times 4 \Rightarrow 20x - 28y = 88$
$\left( 2 \right) \times 5 \Rightarrow 20x - 15y = 75$
Now adding the two equations we obtain an equation,
\[\begin{array}{*{20}{c}}
  {20x - 28y = 88} \\
  {\dfrac{\begin{gathered}
  20x - 15y = 75 \\
   - \,\,\,\,\,\,\, + \,\,\,\,\,\,\,\,\,\, - \\
\end{gathered} }{{0x - 13y = 13}}}
\end{array}\]
Thus we get,
$ \Rightarrow - 13y = 13$
On dividing the term and we get
$ \Rightarrow y = - \dfrac{{13}}{{13}}$
$ \Rightarrow y = - 1$
Now, we substitute $y = - 1$ in any one of the above equation to solve for $x.$
Substituting $y = - 1$ in equation $\left( 1 \right)$
We obtain,
$ \Rightarrow 5x - 7\left( { - 1} \right) = 22$
Let us multiply we get,
$ \Rightarrow 5x + 7 = 22$
On rewriting the term and we get
$ \Rightarrow 5x = 22 - 7$
On subtracting the term and we get,
$ \Rightarrow 5x = 15$
Let us divide 5 on both sides we get
$ \Rightarrow x = \dfrac{{15}}{5}$
On dividing,
$ \Rightarrow x = 3$
Thus, we get
$ \Rightarrow x = 3$

Now, $\left( {3, - 1} \right)$ is a solution to the given system of equation

Note: Here, the two given equations are true when $x = 3$ and $y = - 1$ as from the given two equations.
Substitute $x = 3$ and $y = - 1$ in equation $\left( 1 \right)$ and $\left( 2 \right)$ we get,
$ \Rightarrow 5\left( 3 \right) - 7\left( { - 1} \right) = 22$
$ \Rightarrow 15 + 7 = 22$
Also, $4x - 3y = 15\,$
$ \Rightarrow 4\left( 3 \right) - 3\left( { - 1} \right) = 15$
$ \Rightarrow 12 + 3 = 15\,$
There are three ways to solve systems of linear equations in two variables: graphing, substitution and elimination method.
In the elimination method you either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.
The elimination method for solving systems of linear equations uses the additional property of equality.