Problem 10 - College Algebra Essentials

Question

In Exercises \(1-18,\) solve each system by the substitution method. $$ \left\\{\begin{array}{l} x y=-12 \\ x-2 y+14=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The solutions to the system of equations are (-2, 6) and (-12, 1).

Step 1: Solve First Equation for \(x\)

Start by isolating \(x\) in the first equation. This can be done by dividing every term in the equation by \(y\). This results in \(x = -12/y\).

Step 2: Substitute \(x\) into Second Equation

Now substitute \(x\) as described above into the second equation: \[-12/y - 2y + 14 = 0\]. This equation contains only \(y\), which allows us to solve for \(y\).

Step 3: Solve for \(y\)

Rearrange the equation and solve for \(y\):\[-12/y = 2y - 14 \ -12 = 2y^2 - 14y \ 0 = 2y^2 - 14y + 12. \ Divide by 2 to simplify: \0 = y^2 - 7y + 6. \ Now factor the equation: \ 0 = (y - 6)(y - 1). Setting each factor equal to 0 yields two possible solutions for \(y\): \(y = 6\) or \(y = 1\).

Step 4: Substitute \(y\) into \(x = -12/y\)

Substitute \(y\) into \(x = -12/y\) to solve for \(x\) when \(y = 6\) and \(y = 1\). For \(y = 6\), \(x = -12/6 = -2\). For \(y = 1\), \(x = -12/1 = -12\). The solutions to the system of equations are therefore \((-2, 6)\) and \((-12, 1)\).

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations that share two or more unknowns. The objective is to find values for the variables that satisfy all equations simultaneously.

Typically, the system is written in a format where each equation is aligned, making it easy to visualize the coefficients for each variable. For example, the system given in the exercise:
\[\begin{x}xy=-12 \x-2y+14=0\end{x}\]
Here, we have two equations involving two variables, 'x' and 'y'. The solution to a system like this represents the point(s) where the graphs of these equations intersect when plotted on coordinate axes.

The substitution method, one of the algebraic methods to solve such systems, involves isolating one variable in one equation and then substituting the resulting expression into the other equation(s).

Solving Systems of Equations

Solving systems of equations can be tackled through various strategies. The substitution method is particularly useful when one variable can easily be isolated and then replaced into the other equation.

Here are the key steps involved in solving systems using substitution:

• Isolate a Variable: Choose one of the two equations and express one variable in terms of the other.
• Substitute: Replace the isolated variable in the other equation(s) with the expression obtained in step 1.
• Solve: Solve the resulting equation, which is now in one variable, to find the value of that variable.
• Back-Substitute: Substitute the value found back into one of the original equations to find the value of the other variable.

This process transforms the system from two variables to a single variable equation, allowing for straightforward resolution. In our original exercise, once the value of 'y' is found, it can be plugged back into the original equation to find the value of 'x', providing the complete solution set.

Algebraic Methods

Algebraic methods, like the substitution method used in our example, are analytical approaches to solving systems of equations. They differ from graphical methods, which involve plotting the equations on a graph and identifying the intersection points.

Other than substitution, algebraic methods include elimination (adding or subtracting equations to eliminate one variable) and using matrix operations (employing matrices and determinants). The choice of method often depends on the complexity of the system and the preference of the solver.

In the substitution method, careful manipulation of equations is required to avoid algebraic errors. It's important to check the solutions by substituting them into the original equations to confirm their validity. This method proves to be a powerful tool in algebra, with applications extending to multiple fields, including economics, engineering, and physics.