Question

How do you solve the system of equations $y = - 3x - 12$ and $y = x + 4$ ?

Answer 1

Hint: We solve the given two equations simultaneously. We multiply the second equation with $ + 3$ so that the like terms can be canceled out and on adding the first equation and the resultant equation that we get after multiplying. After we find the value of one variable, we substitute it in the original equation to get the value of the other variable.

Complete step-by-step solution:
In this question, we are asked to solve the given set of equations to find the respective values of $x$ and $y$. The two equations are given as
 $y = - 3x - 12$………… let this be equation (1)
 $y = x + 4$……………. Let this be equation (2)
We will solve the two equations simultaneously. In order to do so, let us multiply equation (2) with $ + 3$ so that the like terms can be canceled out. Thus,
\[ \Rightarrow y = - 3x - 12\]
\[ \Rightarrow 3y = 3x + 12\]
Now let us add the above two equations. Thus we get,
$\begin{array}{*{20}{c}}
  {y = \not{{ - 3x}}\not{{ - 12}}} \\
  {\underline {3y = \not{{ + 3x}}\not{{ + 12}}} } \\
  {4y = 0}
\end{array}$
Therefore, $y = 0$
On substituting this value of $y$ in equation (2), we get:
$ \Rightarrow y = x + 4$
On adding $ - 4$ to both sides, we get:
$ \Rightarrow - 4 = x$

Thus $\left( {x,y} \right) = \left( { - 4,0} \right)$

Additional Information: These kinds of equations are known as Linear Equations. They are also known as the first order equation as they only have variables raised to the power of one. They are the simplest and the most elemental type of equations. These equations are generally defined for lines in a coordinate system. An equation for a straight line is called a linear equation. The equation for a straight line is given as: \[y = mx + c\]
Where ‘m’ represents the slope or the angle formed by the line on the x-axis, and c represents the y-intercept.

Note: Alternate way of solving this question is through the method of substitution.
As we know that,
$y = - 3x - 12$………… let this be equation (1)
 $y = x + 4$……………. Let this be equation (2)
Substituting the value of $y$ given in equation (2) in equation (1), we get:
$ \Rightarrow x + 4 = - 3x - 12$
Keep the variables on the left hand side and the constants on the right hand side, and change their signs accordingly:
$ \Rightarrow x + 3x = - 12 - 4$
$ \Rightarrow 4x = - 16$
Therefore, $x = - 4$
Substituting this value in equation (2), we get:
$ \Rightarrow y = - 4 + 4 = 0$
Thus we have our required answers.