Problem 21 - Algebra 1: Concepts and Skills

Question

Use linear combinations to solve the linear system. Then check your solution. \(2 p-q=2\) \(2 p+3 q=22\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(p = 3.5\) and \(q = 5\).

Step 1: Isolate one variable

In the first equation, isolate p. To do this, add q to both sides of the equation \(2 p-q=2\) to get: \(2p = q+2\). Then, to solve for p, we'd divide by 2 on both sides: \(p= \frac{1}{2}q + 1\).

Step 2: Substitute the value of p

Substitute \(\frac{1}{2}q + 1\) for p into the second equation \(2p + 3q = 22\). This substitution gives us \(2(\frac{1}{2}q + 1) + 3q = 22\). Simplifying this equation gives us: \(q+2+3q=22\), further simplifying gives us \(4q = 20\), therefore \(q=5\).

Step 3: Solve for the other variable

Substitute \(q=5\) into the isolated equation to get the value of p: \(p=\frac{1}{2}*5+1 = 3.5\).

Step 4: Verify the solution

Plugging \(p = 3.5\) and \(q = 5\) into the original system:We have equation 1: \(2p - q = 2*3.5-5 = 2\), which matches.And equation 2: \(2p + 3q = 2*3.5 + 3*5 = 22\), which also matches. Thus the solutions work for both equations.

Key Concepts

Linear Combinations Method

The linear combinations method, also known as the addition method, is one of the approaches used to solve systems of linear equations. This method involves adding or subtracting the equations in the system to eliminate one of the variables, which then allows you to solve for the other.

Let's consider our example where we have a system of two equations:

• \(2p - q = 2\)
• \(2p + 3q = 22\)

To apply the linear combinations method, we need to manipulate the equations such that when we add them together, one variable cancels out. That often involves multiplying one or both equations by a number that will create coefficients in front of the variable that are opposites.

However, in our example, there's no need to multiply as we already have the same coefficient for the variable 'p' in both equations. By subtracting the first equation from the second, we directly obtain an equation with only one variable, 'q', which can then be easily solved. This simplicity is not always the case, and with systems that have coefficients that are not ready for elimination, the process involves more steps.

Systems of Equations

Systems of equations are collections of two or more equations that share a set of unknowns. The goal is to find the values of these unknowns that satisfy all equations in the system simultaneously.

In our example, we are looking at a system with two equations and two unknowns, 'p' and 'q'. Solving such a system means finding a pair of \(p, q\) values that make both equations true. There are different types of systems:

• Consistent systems have one or more solutions.
• Inconsistent systems have no solution.
• Dependent systems have infinitely many solutions usually represented as an equation or parametric form.

Our example is a consistent system as it has a unique solution. Identifying the type of system is vital because it helps determine the most effective method for solving it.

Substitution Method

The substitution method is another technique for solving systems of equations, particularly useful when one equation can be easily solved for one variable in terms of the others. Once we have an expression for this variable, we substitute it into the other equation(s).

Looking back at our problem, in the first step, we isolated 'p' from one equation and got \(p=\frac{1}{2}q + 1\). This expression for 'p' was then substituted into the second equation to find the value of 'q'. Once 'q' was known, that value was used to find 'p' using our isolated equation.

This method is often very straightforward and is particularly advantageous when you can isolate a variable with a coefficient of 1. The main challenge with the substitution method can arise when dealing with fractions or more complex expressions, which can lead to errors if not handled with care.

Algebraic Verification

After solving a system of equations and obtaining the solution, it's crucial to verify that the solution is correct. Algebraic verification involves substituting the answers back into the original equations to ensure that they make each one true.

In our case, we found that \(p = 3.5\) and \(q = 5\). To verify, we plugged these values into the original equations: \(2(3.5) - 5 = 2\) and \(2(3.5) + 3(5) = 22\). Since both equations were satisfied, our solution was verified.

This step is essential to confirm the correctness of the solution. It acts as a proof that the values found algebraically are indeed the solution to the system, and it prevents overlooking any possible arithmetic mistakes made during the solving process.