Problem 32 - Algebra 1

Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &3 p-2=-q\\\ &-q+2 p=3 \end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system of linear equations is \( p = 1 \) and \( q = -1 \).

Step 1: Rewrite the system maintaining the order of variables.

First, rewrite the system of equations maintaining the order of variables. To do so, rewrite \(-q\) as \(q\) on the left side. The system becomes: \[ \begin{aligned} &3p - q=2 \ &2p + q=3 \end{aligned} \]

Step 2: Apply the linear combination.

Second, add the two equations together to eliminate 'q'. Performing \( (3p - q) + (2p + q) \) gives \( 5p = 5 \).

Step 3: Solve for p.

Third, solve for 'p' by dividing both sides by 5. It gives \( p = 1 \).

Step 4: Substitute p into the first original equation.

Fourth, replace 'p' with '\(1\)' into the first original equation \(3p - q = 2\), hence \(3(1) - q = 2\), which simplifies as \(3 - q = 2\).

Step 5: Solve for q.

Finally, isolate 'q' on the left side by following these steps: Subtract '3' from both sides to get \(-q = 2 - 3\), then multiply each side by '-1' to get '\(q = -1\)'.

Key Concepts

System of Linear Equations

A system of linear equations is a set of two or more linear equations that share common variables. In our problem, the system is defined by the variables \(p\) and \(q\). Understanding how these equations work is fundamental:

• Each equation in the system represents a straight line when graphed on a coordinate plane.
• Solving the system means finding a point where the equations intersect, which gives the values of \(p\) and \(q\).
• Both equations need to be true at the same time for these variable values.

In our exercise, we start with two linear equations. By manipulating them, we look for a common solution to both. This is where techniques like solving equations and elimination methods become invaluable.

Solving Equations

When we solve a system of linear equations, our goal is to find values for the variables that satisfy all equations simultaneously. Here are some key steps involved:

• Rearrange the equations if necessary to align terms neatly.
• Identify the easiest entry point, i.e., the variable or equation that can be handled first.
• Apply algebraic methods like adding, subtracting, or substituting values wherever needed.

In the given problem, rearranging the equations ensured clarity for further steps. Solving one equation led us to eliminate one variable, simplifying everything to a single variable equation.

Elimination Method

The elimination method is a hallmark of solving systems of linear equations, especially when variables align conveniently for cancellation. Here’s how it works:

• Adjust one or both equations if necessary so that when added, one variable cancels out.
• Perform arithmetic operations such as addition or subtraction on the entire equations.
• This leaves a simpler equation with just one variable, making it easier to solve.

In the original exercise, we effectively used elimination by adding the two revised equations. This removed the \(q\) variable, allowing us to solve for \(p\) directly.

Algebra

Algebra is the mathematical language used to express relationships and solve equations. It allows us to move beyond basic arithmetic:

• Variable representation simplifies the description of equations.
• We often use algebra to rearrange equations, isolate variables, and find solutions.
• Techniques like distribution, factoring, and combining like terms are essential tools.

In solving our problem, algebraic manipulation, such as distributing terms and neat rearrangement, plays a crucial role. Whether isolating a variable or finding common denominators for elimination, algebra provides structure and pathways to solutions.