Problem 46 - Elementary Algebra Within Reach

Question

Write a system of linear equations that is more efficiently solved by the method of elimination than by the method of substitution. (There are many correct answers.)

Step-by-Step Solution

Answer

A possible example of a system of linear equations, which is easier to solve using the method of elimination, could be: \(3x + 2y = 6\) and \(3x - y = 1\).

Step 1: Formulation of the first equation

Choose the first equation. For instance, let the equation be \(3x + 2y = 6\).

Step 2: Formulation of the second equation

Formulate the second equation making sure that the coefficients of one of the variables are equal in both equations. For instance, the second equation can be \(3x - y = 1\) where the coefficients of x in both equations are equal.

Step 3: Verification of the system of equations

The system of the equations now is: \[3x + 2y = 6\] \[3x - y = 1\] This system of equations will be more efficiently solved by the method of elimination because one can simply subtract the second equation from the first to eliminate the variable x.

Key Concepts

System of Linear Equations

In algebra, a system of linear equations consists of two or more linear equations that have the same set of variables. The goal is to find a solution that satisfies all equations simultaneously, which means finding the values of the variables that make all the equations true.

These systems can often represent real-world problems, such as predicting expenses, solving geometry problems, or finding the intersection of two lines. To further illustrate, consider a scenario where you're budgeting for a party with two types of foods, and you know the total cost and the total items you need to buy. This can be represented with a system of linear equations: one for the cost and another for the number of items. Solving the system will tell you how much of each type of food you can afford within your budget.

To solve a system of linear equations, we can use graphic methods by drawing lines on a graph to identify points of intersection or algebraic methods such as substitution, elimination, or matrix operations.

Elimination vs Substitution

When tackling a system of linear equations, two primary algebraic methods to consider are elimination and substitution.

Substitution is a method where we solve one of the equations for one variable and then substitute that expression into the other equations. This method is best used when one equation can be easily solved for one variable, making substitution straightforward.

Elimination, on the other hand, involves adding or subtracting equations to eliminate one of the variables. It allows us to combine the equations in a way that results in an equation with just one variable. This method is efficient if you can quickly make the coefficients of one variable the same or exact opposites, which leads to that variable canceling out when the equations are added or subtracted.

Solving Linear Equations

To solve linear equations, regardless of the method chosen, we must manipulate the equations until all variables are isolated. Let's focus on the method of elimination, which works best when variables can be easily eliminated through addition or subtraction of the equations.

In practice, here are the steps typically followed:

1. Align the equations with matching variables on top of each other.
2. Adjust the equations by multiplication or division, if necessary, to obtain opposite coefficients for a variable.
3. Add or subtract the equations to eliminate that variable.
4. Solve the resulting equation for the remaining variable.
5. Substitute the found value into one of the original equations to solve for the other variable, if there are only two variables to solve for.
6. Check the solution by plugging the values back into the original equations to ensure they hold true.

These steps should result in the values that solve the original system of equations.

Algebraic Methods

Several algebraic methods exist to solve equations, with each suited to different types of problems. Aside from substitution and elimination, we also have the graphical method, where we plot the equations on a graph and look for the intersection point(s) to find the solutions. Another powerful tool is using matrices, where systems of equations are converted into a matrix form and solved using matrix operations.

In the context of the exercise provided, the elimination method is ideal because the coefficients of one variable (x) are the same in both equations, meaning we can subtract one equation from the other directly to find the value of y. It is also less error-prone and generally faster when the equations are structured for easy elimination, as seen in our step-by-step solution. Remember, the key is to identify the method that simplifies the process of finding the solution in the most effective way possible.