Problem 77 - Precalculus

Question

The point (2,-3) is a solution to the system of equations $$ \begin{array}{l} A x+B y=-29 \\ A x-B y=13 \end{array} $$ Find \(A\) and \(B\)

Step-by-Step Solution

Answer

The values are \(A = -4\) and \(B = 7\).

Step 1: Substitute Known Values into Equation 1

Start by substituting the coordinates of the point \((2, -3)\) into the first equation of the system. This gives us: \[ A(2) + B(-3) = -29 \] which simplifies to: \[ 2A - 3B = -29 \]

Step 2: Substitute Known Values into Equation 2

Next, substitute the point \((2, -3)\) into the second equation: \[ A(2) - B(-3) = 13 \] This simplifies to: \[ 2A + 3B = 13 \]

Step 3: Solve the System of Equations Using Elimination

We now have two equations: 1. \(2A - 3B = -29\)2. \(2A + 3B = 13\) To eliminate \(A\), subtract Equation 1 from Equation 2: \[(2A + 3B) - (2A - 3B) = 13 - (-29)\] This simplifies to: \[6B = 42\] Solving for \(B\), we get: \[B = 7\]

Step 4: Substitute Back to Find A

Use the value of \(B = 7\) in either of the original equations, let's use the first equation: \[2A - 3(7) = -29\] Simplify: \[2A - 21 = -29\] Add 21 to both sides: \[2A = -8\] Finally, divide by 2: \[A = -4\]

Key Concepts

Substitution Method

The substitution method is a powerful technique used to solve systems of equations, especially when one equation can be easily rearranged to represent one variable in terms of the others. In this exercise, however, substitution isn’t used directly, but it’s important to know how this method works. If you have two equations, such as:

• Equation 1: \(x = 2y + 1\)
• Equation 2: \(3x - y = 7\)

You can take the expression for \(x\) from Equation 1 and substitute it into Equation 2. This means wherever you see \(x\) in Equation 2, you replace it with \(2y + 1\). This way, you reduce the system to a single equation in

one variable, allowing you to solve for \(y\), then back-substitute to find \(x\).

Even though our original exercise was solved with the elimination method, understanding substitution can be very useful for different scenarios where one equation is already isolated.

Elimination Method

The elimination method is all about removing one variable so you can solve for the other. In our original exercise, this method was key. Here’s how elimination works:

• First, make sure both equations are aligned and comparable. This means your variables should line up when written under each other.
• Next, either add or subtract the equations to cancel out one of the variables.

In our exercise, we had:

• Equation 1: \(2A - 3B = -29\)
• Equation 2: \(2A + 3B = 13\)

By subtracting Equation 1 from Equation 2, the \(2A\) terms cancel each other out. This leads to the cancellation of \(A\) in both equations, leaving us with a simpler equation: \(6B = 42\). From here, solving for \(B\) becomes straightforward: you just divide by 6 to get \(B = 7\).
Finally, substitute this value back into either of the original equations to find \(A\). This strategy helps turn two equations into manageable single-variable problems.

Solving Linear Equations

Solving linear equations is at the heart of many algebra problems. A linear equation is any equation that forms a straight line when graphed. The general form is \(ax + by = c\). The key steps to solving a set of linear equations are systematic.

• First, simplify each equation if necessary, which means clearing out parentheses and combining like terms.
• Decide on a method to use: either substitution, elimination, or graphing.
• Solve for one of the variables, and use the results to find the other unknowns.

In this exercise, we didn't need to graph because the orientation and tasks were manageable with elimination. Solving linear equations using algebraic methods like substitution or elimination is effective for systems involving two variables. In more complex systems, keeping a structured approach ensures logical steps and correct solutions.
Understanding these principles doesn’t just help in math class; it sharpens logical thinking and problem-solving skills applicable in everyday situations.