Problem 8 - Algebra 2

Question

Solve each system of equations by using elimination. \(2 p+4 q=18\) \(3 p-6 q=3\)

Step-by-Step Solution

Answer

The solution is \(p = 5\) and \(q = 2\).

Step 1: Setup the System of Equations

We start with the given system of equations:1. \(2p + 4q = 18\)2. \(3p - 6q = 3\). We need to eliminate one of the variables by making their coefficients opposites.

Step 2: Align the Coefficients

To eliminate \(q\), we can multiply the first equation by 3 and the second equation by 2 to align the coefficients of \(q\) as opposites:\[ \begin{align*}3(2p + 4q) &= 3(18) \2(3p - 6q) &= 2(3)\end{align*} \]This yields:1. \(6p + 12q = 54\)2. \(6p - 12q = 6\).

Step 3: Add the Equations

Add the two equations to eliminate \(q\):\[ (6p + 12q) + (6p - 12q) = 54 + 6 \]This simplifies to:\[ 12p = 60 \].

Step 4: Solve for the Remaining Variable

Solve for \(p\) by dividing both sides of the equation by 12:\[ p = \frac{60}{12} = 5 \].

Step 5: Substitute to Find the Other Variable

Substitute \(p = 5\) back into one of the original equations to solve for \(q\). Using the first equation:\[ 2(5) + 4q = 18 \]\[ 10 + 4q = 18 \]Subtract 10 from both sides:\[ 4q = 8 \]Divide by 4:\[ q = \frac{8}{4} = 2 \].

Step 6: Verify the Solution

Verify the solution \((p, q) = (5, 2)\) by plugging it into the second equation:\[ 3(5) - 6(2) = 3 \]\[ 15 - 12 = 3 \]Since it holds true, our solution is correct.

Key Concepts

Understanding a System of Equations

A system of equations is essentially a set of two or more equations with multiple variables. Each equation in the set provides information about how the variables are related. The main goal when dealing with such a system is to find the values of the variables that satisfy all the equations at the same time.

Imagine you're a detective, and each equation is a clue. You need to find the exact values that solve the mystery, which in this case is the correct values for the variables. A typical example is solving two equations with two variables such as the ones in our exercise: \(2p + 4q = 18\) and \(3p - 6q = 3\). Each equation on its own is a straight line when graphed, and the solution is the point where these two lines intersect.

In mathematics, these solutions are often found using methods like substitution, graphing, or, as we use here, elimination. Each method offers a different way to look at the relationships and ultimately solves the problem of finding that intersection point — the solution! The key takeaway here is that a system of equations allows you to explore relationships between multiple variables.

Exploring Variable Elimination

Variable elimination is a method used to remove one of the variables in a system of equations so that the rest can be solved more easily. It's a common technique that simplifies the problem by focusing on one variable at a time. This method works by aligning the coefficients of the variable you wish to eliminate and then adding or subtracting the equations to cancel it out.

In our exercise, we aimed to eliminate the variable \(q\). To do this, we adjusted the equations so that the coefficients in front of \(q\) were opposite numbers. We multiplied the first equation by 3 and the second equation by 2. This gave us new equations: \(6p + 12q = 54\) and \(6p - 12q = 6\). By adding these equations together, \(q\) was successfully eliminated, leaving us with an equation in just one variable: \(12p = 60\).

The elimination method is highly efficient as it reduces the complexity of dealing with multiple equations. When applied correctly, it leads us straight to the answer for one of the variables, which can then be used to find the other variable. This approach is particularly useful for linear equations, where variables are combined in simple, predictable ways.

Breaking Down Linear Equations

Linear equations are a basic form of algebraic expressions where each term is either a constant or the product of a constant and a single variable. The key feature of linear equations is their simplicity and direct nature, making them the cornerstone of many mathematical concepts.

For example, the equation \(2p + 4q = 18\) is a linear equation because it doesn't involve any powers or complicated operations on the variables \(p\) and \(q\). Each variable is raised to the power of 1, keeping the equation as straightforward as possible. When represented graphically, linear equations form straight lines, and our task in solving a system of such equations is to find the intersection of these lines.

Understanding linear equations helps in various ways:

• They form the foundation for understanding more complex equations and systems.
• They are used in numerous real-life applications, from calculating costs to predicting trends.
• They offer a clear pathway for applying algebraic techniques like substitution and elimination.

Being adept at managing linear equations ensures you can handle a range of mathematical problems with ease.