Problem 24 - College Algebra Essentials

Question

In Exercises \(19-30,\) solve each system by the addition method. $$ \left\\{\begin{array}{l} 2 x-7 y=2 \\ 3 x+y=-20 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = -6\) and \(y = -2\).

Step 1: Equalize Coefficients

First, aim to match the coefficients of 'y' in both equations. Notice the coefficients of y (i.e. -7 and 1) can be made identical by multiplying the second equation by 7. It results in: \(2x - 7y = 2\) (unchanged first equation) and \(21x + 7y = -140\) (second equation after multiplication).

Step 2: Add the Equations

Add the two equations in order to cancel out the y variable, leading to: \(2x + 21x = 2 - 140\), which simplifies to \(23x = -138.\)

Step 3: Solve for x

Now, solve for the x variable. By dividing both sides by 23, x can be isolated: \(x = -138/23, x = -6.\)

Step 4: Substitute x into the First Equation

Next, substitute x = -6 into the first equation of the system: \(2(-6) - 7y = 2\), simplifying to \(-12 - 7y = 2\).

Step 5: Solve for y

Then, solve for y by adding 12 to both sides and dividing by -7: \(-7y = 14, y = -2.\)

Step 6: Conclusion

Therefore, the solution to the system of equations is \(x = -6\) and \(y = -2\).

Key Concepts

System of Equations

A system of equations is a set of two or more equations that share the same set of variables. In the exercise you encountered, there are two equations and two variables (x and y). This is an example of a linear system. The goal when working with systems of equations is to find a set of values for the variables that satisfies all the equations simultaneously.

Consider the example system:

• Equation 1: \(2x - 7y = 2\)
• Equation 2: \(3x + y = -20\)

These equations form a system. To solve it, you need to find values for \(x\) and \(y\) that make both equations true. This process is essential in various fields, including science, engineering, and economics, where systems of equations are used to model real-world situations.

Solving Linear Equations

Solving linear equations is a method to determine the values of variables that satisfy an equation. In the addition method, also known as the elimination method, you aim to eliminate one variable by adding or subtracting equations. This is particularly useful for systems with two equations.

For the provided exercise, the first step is to make the coefficients of one variable (in this case, \(y\)) equal in both equations:

• Multiply the second equation by a number that matches the coefficients of \(y\) in the first equation.
• In this example, multiply the second equation by 7 to get the same magnitude of coefficient for \(y\): \(3x + y = -20 \rightarrow 21x + 7y = -140\).
• By adding this new equation to the first, you eliminate \(y\): \(2x - 7y = 2\) and \(21x + 7y = -140\).

Once \(y\) is eliminated, solving for \(x\) becomes straightforward.
Finally, use the value of \(x\) obtained to solve for \(y\) by substituting \(x\) back into one of the original equations. This two-step process ensures all variables are accounted for.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vectors, vector spaces, and linear equations. A key part of linear algebra involves solving systems of linear equations, just like the system you worked on in the given exercise. Linear algebra provides tools and methods to understand relationships between multiple variables, often represented as matrices or equations.

Understanding systems of equations is crucial in linear algebra because:

• Their solutions can often be visualized as intersections of lines, planes, or hyperplanes in geometry.
• They provide insights into multi-dimensional data structures in data science and computer graphics.

Being proficient at solving systems of linear equations enhances your problem-solving capabilities in theoretical and applied disciplines. By mastering such methods, you can tackle more complex problems, from predicting trends with statistical models to optimizing resource allocations in operations research.