Solve the system of equations:
$\begin{aligned}-4 x-3 y & =-10 \\ 4 x-4 y & =24\end{aligned}$
The system of linear equations is given by:
(1) \(-4x - 3y = -10\)
(2) \(4x - 4y = 24\)
We have two linear equations with two variables, \(x\) and \(y\). Since the coefficients of \(x\) are \(-4\) and \(4\), they are additive inverses. This makes the elimination method the most efficient way to solve the system.
To eliminate \(x\), we add equation (1) and equation (2) together:
\((-4x - 3y) + (4x - 4y) = -10 + 24\)
The \(x\) terms cancel out:
\((-4x + 4x) + (-3y - 4y) = 14\)
\(-7y = 14\)
Now, solve for \(y\):
\(y = \frac{14}{-7} = -2\)
Now that we have the value \(y = -2\), we can substitute it into one of the original equations to find \(x\). Let's use equation (2):
\(4x - 4y = 24\)
Substitute \(y = -2\):
\(4x - 4(-2) = 24\)
\(4x + 8 = 24\)
Subtract 8 from both sides:
\(4x = 24 - 8\)
\(4x = 16\)
Divide by 4:
\(x = \frac{16}{4} = 4\)
We have found \(x = 4\) and \(y = -2\). We can verify this by plugging the values into equation (1):
\(-4(4) - 3(-2) = -16 + 6 = -10\).
The values satisfy both equations. Thus, the final solution is \(x = 4\) and \(y = -2\), expressed as the coordinate pair \((4, -2)\).
Systems of linear equations are used in various real-life scenarios, such as determining the intersection point of two paths. For example, if two companies offer different pricing structures for a service—Company A charges a flat fee plus a rate per hour, and Company B charges a different flat fee plus a different rate—you can model these as linear equations. Solving the system allows you to find the number of hours at which the cost for both companies is equal, helping you decide which option is more cost-effective.