Problem 20 - Algebra Form and Function

Question

Solve the systems of equations. $$ \left\\{\begin{array}{l} 4 p-7 q=2 \\ 5 q-3 p=-1 \end{array}\right. $$

Step-by-Step Solution

Answer

Answer: The values of p and q that satisfy the given system of linear equations are p = -3 and q = -2.

Step 1: Multiply the equations to eliminate one variable

Multiply the first equation by 3 and the second equation by 4 to make the coefficients of p the same in both equations. This will allow us to eliminate p by adding the two equations: (1) \(3(4p - 7q) = 3(2)\). (2) \(4(5q - 3p) = 4(-1)\).

Step 2: Solve the modified equations

After multiplying the equations: (1) \(12p - 21q = 6\). (2) \(-12p + 20q = -4\).

Step 3: Add the modified equations so as to eliminate p

Add equation (1) and (2) to eliminate p: \((12p - 21q) + (-12p + 20q) = 6 + (-4)\). Hence, \(-1q = 2\).

Step 4: Solve for q

Divide the equation by \(-1\): \(q = -2\).

Step 5: Substitute the value of q into one of the original equations to solve for p

We can use the first original equation to solve for p: \(4p - 7(-2) = 2\). Now, simplify and solve for p: \(4p + 14 = 2\) \(4p = -12\) \(p = -3\)

Step 6: Write the solution

The solution for the system of linear equations is: \(p = -3\) \(q = -2\).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form a straight line when graphed on a coordinate plane. They commonly appear in the format of ax + by = c, where x and y are variables, and a, b, and c are constants. These equations can describe real-world situations such as determining the cost of items or predicting trends. In this exercise, we dealt with a system of linear equations:

• 4p - 7q = 2
• 5q - 3p = -1

These equations contain the variables p and q, and our goal was to find the values that satisfy both equations simultaneously. Systems of linear equations can be represented in many ways, such as graphs, tables, or algebraic forms.
Understanding linear equations is crucial since they are foundational in both algebra and all higher mathematics.

Elimination Method

The elimination method is a systematic approach to solving systems of equations, where we "eliminate" one variable to make it easier to solve for the other. This method is particularly useful when the equations are already in standard form and allows for a streamlined calculation process.
In the given system of equations, we employed the elimination method by aligning the coefficients of "p" so they could cancel each other out when the equations are added:

• First, multiply the first equation by 3: 12p - 21q = 6.
• Then, multiply the second equation by 4: -12p + 20q = -4.

Once we've set it up, we can add these to eliminate the variable p, making the problem simpler. The elimination method shines in its ability to reduce complications in equations by focusing on just one variable at a time.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. In this exercise, the solution to the system of linear equations was found by eliminating one variable and solving step-by-step.
After using the elimination method to remove p, we found the following simplified equation:

• -1q = 2

From here, the process is straightforward. Divide by -1 to isolate q:

• q = -2

Next, substitute this value back into one of the original equations to find the corresponding value of p:

• 4p - 7(-2) = 2 leads to p = -3

Each step involves simplifying an equation, making it easier to solve. The key is to remain methodical, ensuring that each step logically follows from the previous one, ultimately arriving at a solution that satisfies both equations.