Problem 27 - Algebra 1: Concepts and Skills

Question

Use linear combinations to solve the linear system. Then check your solution. \(2 q=7-5 p\) \(4 p-16=q\)

Step-by-Step Solution

Answer

The solution to the system of equations is \(p = 3\) and \(q = -4\).

Step 1: Rearrange First Equation

In the first equation, isolate \(q\) to find an equation that can be used for substitution in the second equation. \(2q = 7 - 5p\) can be written as \(q = (7 - 5p)/2\).

Step 2: Substitute first Equation into Second Equation

Next substitute \(q\) from first equation into second equation, \(4p - 16 = (7 - 5p)/2\). Multiply equation by 2 to clear the denominator getting \(8p - 32 = 7 - 5p\).

Step 3: Solve for p

Re-arrange the equation to isolate \(p\), \(8p + 5p = 32 + 7\). That simplifies to \(13p = 39\). Divide both sides by 13, to obtain \(p = 39/13\), so \(p = 3\).

Step 4: Solve for q

Next, substitute \(p = 3\) into the first equation to solve for \(q\), \(2q = 7 - 5*3\). That simplifies to \(2q = 7 - 15\), then \(2q = -8\). By dividing by 2 on each side, obtain \(q = -8/2\), so \(q = -4\).

Step 5: Verify the Solution

Finally, verify the solution by substituting \(p = 3\) and \(q = -4\) into the original equations. Doing so confirms that \(p = 3\) and \(q = -4\) are the valid solutions to the system of equations.

Key Concepts

Linear Combinations

Linear combinations are a powerful technique used to solve systems of linear equations. This method involves adding or subtracting equations to eliminate a variable, making it easier to find the solution. In this process, we often manipulate the equations to align coefficients for convenient elimination. For example, if you have an equation like \(a_1x + b_1y = c_1\) and another equation \(a_2x + b_2y = c_2\), you might multiply each equation by a suitable number to ensure that the coefficients of either \(x\) or \(y\) in both equations match. After this, simply add or subtract the equations, which cancels out one of the variables, leaving a single-variable equation to solve.

• Align coefficients
• Add or subtract equations
• Eliminate a variable

This technique is extremely useful, especially when working with more complex systems where substitution and graphical methods may not be feasible.

System of Equations

A system of equations is defined as a set of two or more equations with the same variables. For example, consider our original system:

• \(2q = 7 - 5p\)
• \(4p - 16 = q\)

The goal is to find values for the variables \(p\) and \(q\) that satisfy both of these equations simultaneously. These values make up the solution of the system. The methods used to solve systems of equations include substitution, elimination, and graphing. Each method has its own advantages depending on the specific problem at hand.
For linear systems with two variables, like our example, graphical methods involve plotting both equations on a coordinate grid to find the intersection point. However, algebraic methods like substitution or linear combinations are more precise and effective for larger systems or when the equations don't easily graph. Understanding how to correctly identify the type of system and choose the right method to solve it is a fundamental skill in algebra.

Substitution Method

The substitution method is a strategic approach to solving a system of equations. Here’s a simple breakdown of this technique:

• Start by solving one of the equations for one variable in terms of the other.
• Substitute this expression into the other equation.
• Solve the resulting equation for the single variable.
• Use the value of this variable to find the value of the other variable by plugging back into one of the initial equations.

In the given problem, after rearranging the first equation, we express \(q\) in terms of \(p\) as \(q = \frac{7 - 5p}{2}\). This expression is then substituted into the second equation, effectively reducing the number of variables and simplifying the process of solving the equation set.
Substitution is particularly useful when one of the equations is easily solvable for a single variable. It allows you to handle systems with potentially less complexity and greater clarity.

Equation Solving

Equation solving is fundamentally about finding the variable values that make the equation true. When solving any linear equation, it is important to isolate the variable on one side of the equation. In our example, once the substitution is done, we arrive at an equation like \(8p - 32 = 7 - 5p\). Solving this involves basic algebra:

• Combine like terms
• Isolate the variable
• Perform arithmetic operations

Initially, we combine the terms related to \(p\) and move constant terms to one side. This might form an equation resembling \(13p = 39\). Dividing both sides by the numerical coefficient of \(p\) solves for \(p\). After finding \(p\), back-substitute to find \(q\). Equation solving is about following these steps logically to ensure each has been correctly applied, eventually leading you to the precise values that satisfy the original equations.