Problem 176 - Elementary Algebra

Question

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination. (a) \(\left\\{\begin{array}{l}y=7 x-5 \\ 3 x-2 y=16\end{array}\right.\) (b) \(\left\\{\begin{array}{l}12 x-5 y=-42 \\ 3 x+7 y=-15\end{array}\right.\)

Step-by-Step Solution

Answer

Use substitution for part (a) and elimination for part (b).

Step 1 - Identify the forms of the equations

First, look at both systems of equations to identify their forms. For part (a), the first equation is already solved for y. For part (b), neither equation is solved for a single variable yet.

Step 2 - Decide on the method for part (a)

In part (a), since the first equation is already solved for y (\( y = 7x - 5 \)), substitution is the more convenient method. Substitute \( y \) from the first equation into the second equation.

Step 3 - Decide on the method for part (b)

In part (b), neither equation is solved for a single variable. Observing the coefficients, elimination would be more convenient because the x-terms or y-terms can be manipulated to cancel each other out. Specifically, multiplying the first equation by \(3\) and the second equation by \(12\) can help eliminate one of the variables easier.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of linear equations by solving one of the equations for one variable, and then substituting this expression into the other equation. This method works particularly well when one equation is already solved for one of the variables.
In step 1 of the original exercise above, part (a) has the equation \(y = 7x - 5\), which is already solved for \(y\). Thus, we can use substitution easily here by inserting \(7x - 5\) for \(y\) in the second equation.
Steps to use the substitution method:

• Solve one of the equations for one of the variables, if not already done.
• Substitute this expression into the other equation.
• Solve the resulting equation for the single variable.
• Substitute back to find the other variable.

So for part (a) from above:
1. The first equation is already solved: \(y = 7x - 5\)
2. Substitute into the second equation: \(3x - 2(7x - 5) = 16\)
3. Simplify and solve for \(x\): \(3x - 14x + 10 = 16\) leading to \( -11x = 6 \rightarrow x = -\frac{6}{11}\)
4. Substitute \(x\) value back into the first equation to find \(y\): \(y = 7(-\frac{6}{11}) - 5 = -\frac{42}{11} - \frac{55}{11} = -\frac{97}{11}\). This gives the solution \(x = -\frac{6}{11}\), \(y = -\frac{97}{11}\).
This method simplifies the process when an equation is in a suitable form already.

Elimination Method

The elimination method, also known as the addition method, involves adding or subtracting the equations in a system to eliminate one of the variables. This method is useful when neither equation is readily solved for one of the variables.
In step 3 of the original exercise above, part (b) involves equations: \(12x - 5y = -42\) and \(3x + 7y = -15\). Here, neither variable is isolated. However, we can manipulate the equations to cancel out one of the variables.
Steps to use the elimination method:

• Multiply the equations by suitable values if needed so that when added or subtracted, one variable cancels out.
• Add or subtract the equations to eliminate that variable.
• Solve the resulting equation for the remaining variable.
• Substitute this value back into either of the original equations to find the other variable.

For part (b) from above:
1. We need to make the coefficients of either \(x\) or \(y\) equal: Multiply the first equation by \(3\) and the second equation by \(12\): \br \begin{align*}36x - 15y = -126 \ \ (12x - 5y)\times 3\

& \ (3x + 7y)\times 12 \ 36x + 84y = -180 \end{align*}
2. Now subtract the second modified equation from the first: \br
\begin{align*} 36x - 15y - (36x + 84y) = -126 - (-180) \ -99y = 54 \ y = -\frac{54}{99} = -\frac{6}{11} \ \end{align*}
3. Substitute \(y\) value back into one of the original equations to find \(x\):\begin{align*} \[\begin{equation}\ 12x - 5\left(-\frac{6}{11}\right) = -42 \ x = -\frac{41}{9}. \end{equation}\]\end This gives the solution \(x = \frac{57}{41}\) and \( y = - \frac{26}{23}\).

Equation Solving Strategies

Selecting the best method to solve a system of equations depends on the form of the equations in the system. Let’s consider a few strategies:

• If a system includes an equation already solved for one variable, the substitution method is usually more straightforward.
• If both equations are in standard form and neither is solved for a single variable, consider the elimination method, especially if manipulating coefficients can make one variable cancel out with simple addition or subtraction.
• If the system is non-linear or involves higher degree polynomials, other methods like graphing or matrix operations might be more suitable.

For example, in part (a), where one equation is already solved for \(y\), substitution made the solution easier. Conversely, in part (b), using elimination simplfied the process by cancelling one variable and solving a simpler equation.
Understanding the structure of the given system of equations is crucial. Identify if isolating one variable or adding equations will simplify the solution the most. Always check your final solution by plugging the values back into the original equations to ensure accuracy.
Practicing these methods will improve your ability to choose the most efficient strategy for different types of systems.