Problem 18 - Elementary Algebra

Question

Solve each system by using either the substitution or the elimination-by- addition method, whichever seems more appropriate. $$\left(\begin{array}{l}4 x-7 y=21 \\ -4 x+3 y=-9\end{array}\right)$$

Step-by-Step Solution

Answer

The solution is \( x = 0 \) and \( y = -3 \).

Step 1: Choose a Method

To solve this system, we'll use the elimination method because the coefficients of the variable \( x \) are already set to be equal and opposite. This will allow us to eliminate \( x \) easily by adding the two equations.

Step 2: Add the Equations

Add the two equations together to eliminate \( x \):\((4x - 7y) + (-4x + 3y) = 21 + (-9)\)This simplifies to:\(-4y = 12\).

Step 3: Solve for y

Divide both sides of the equation by -4 to solve for \( y \): \(y = \frac{12}{-4} = -3\).Thus, \( y = -3 \).

Step 4: Substitute y into an Original Equation

Now that we have \( y \), substitute \( y = -3 \) into the first original equation to solve for \( x \):\(4x - 7(-3) = 21\).Simplifying gives:\(4x + 21 = 21\).

Step 5: Solve for x

Subtract 21 from both sides to solve for \( x \): \(4x = 0\).Divide by 4 to find \( x = 0 \).

Step 6: Conclusion

The solution to the system of equations is \( x = 0 \) and \( y = -3 \). This means the lines intersect at the point (0, -3).

Key Concepts

Elimination Method

The Elimination Method is a powerful technique for solving systems of equations, especially when working with linear equations. The goal is to remove, or eliminate, one of the variables to simplify the process of solving for the others. This approach is particularly handy when the coefficients of one variable are additive inverses, like in our given system where the coefficients of \( x \) are \( 4 \) and \( -4 \).

The elimination process involves a few key steps:

• Ensure one of the variables can be eliminated easily by adding or subtracting the equations.
• Add or subtract the equations. In our example, we added them: \((4x - 7y) + (-4x + 3y)\). This eliminated \( x \) because \( 4x + (-4x) = 0 \).
• Solve the resulting equation for the remaining variable.

In this case, we ended up with \(-4y = 12\), which simplifies to \( y = -3 \). With \( y \) known, it's straightforward to find \( x \) by substituting back into one of the original equations.

The elimination method is often preferred when dealing with equations that are naturally set up for elimination, saving time and reducing potential errors.

Substitution Method

The Substitution Method is another common strategy for solving systems of equations. This method is particularly useful when one of the equations is already solved for a single variable. It involves substituting this expression into the other equation.

While we used the elimination method for solving the given system of equations, understanding substitution is still crucial:

• Solve one equation for one of the variables, if it's not done already.
• Substitute this expression into the other equation, replacing the variable.
• Solve the new equation for the remaining unknown.
• Substitute this solution back to find the other variable.

Let's explore this with a simple example: Suppose you have \( y = 2x + 5 \) and another equation \( x + y = 10 \). You substitute \( 2x + 5 \) for \( y \) in the second equation to get \( x + (2x + 5) = 10 \). Solving this will give the value of \( x \), which you can use to find \( y \).

The substitution method works well when algebraic manipulation is simpler than elimination, offering flexibility in tackling different systems of equations.

System of Equations

A System of Equations is a set of two or more equations with the same variables. The goal is to find a common solution that satisfies all equations simultaneously. Consider the system in our problem:

\[ 4x - 7y = 21 \]
\[-4x + 3y = -9 \]

Finding the solution involves determining the values of \( x \) and \( y \) that make both equations true. There are several methods to solve these systems, mainly substitution and elimination.

Characteristics of systems:

• If a solution exists, the system is considered consistent.
• It can have one solution (the lines intersect at one point), infinitely many solutions (the lines coincide), or no solution (the lines are parallel).

The given system has a single solution at \( (0, -3) \), indicating the lines intersect at that point. Each method (elimination or substitution) boils down to Algebraic Manipulation to simplify the equations into a solvable form.

Understanding systems of equations is fundamental to solving problems in various fields like engineering, science, and economics, where multiple constraints exist.