Problem 34 - Elementary Algebra Within Reach

Question

Explain how you can check the solution of a system of linear equations algebraically.

Step-by-Step Solution

Answer

To check the solution of a system of linear equations algebraically, substitute the values of the variables into both original equations. If the equations hold true, then the solution is correct.

Step 1: Understand the system of linear equations

Make sure you understand each equation in the system. They should be in the form \(ax + by = c\), where a, b, and c are constants and x, y are variables.

Step 2: Solve the system algebraically

Choose either substitution or elimination method. If you're using substitution, solve one equation for one variable and then substitute that expression into the other equation. If you're using elimination, add or subtract the equations to eliminate one variable, making it possible to solve for the other.

Step 3: Check the solution

Substitute the solution (the values of x, y) into both equations in the original system. If the left-hand side equal to the right-hand side for both equations, then the solution is correct.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. To represent real-life scenarios mathematically, we often use these equations. Each equation in the system has the general form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are coefficients or constants, and \(x\) and \(y\) are the variables we are solving for. When working with such systems, our goal is typically to find the values of \(x\) and \(y\) that satisfy all equations simultaneously. A solution to the system means that the values of \(x\) and \(y\) make both (or all) equations true.

Substitution Method

The substitution method involves solving one of the equations for one variable and then replacing that variable in the other equation with the solved value. This method is particularly useful when one equation can be easily rearranged to isolate one variable. For example, if you have the equation \(x + 2y = 7\), you can solve for \(x\) to get \(x = 7 - 2y\). You then substitute \(7 - 2y\) for \(x\) in the other equation of the system. This leads to an equation in one variable, which can be solved. Once you find the value of one variable, you can substitute it back into any of the original equations to find the value of the other variable.

Elimination Method

The elimination method requires you to add or subtract the equations in the system to eliminate one of the variables. For instance, if one equation is \(2x + 3y = 5\) and the other is \(4x - 3y = 11\), you can add these equations together to eliminate \(y\) because adding \(3y\) to \(-3y\) results in zero, leaving only an equation with \(x\). After finding the value of \(x\), you can substitute it into any original equation to solve for \(y\). This method is efficient when the coefficients of one variable are the same or opposite in both equations, as it allows for quick cancellation of one of the variables.

Algebraic Solution Verification

After solving a system of linear equations using either the substitution or elimination method, it's crucial to verify that the solution is correct algebraically. This involves substituting the solution – the values of \(x\) and \(y\) – back into the original equations. If both equations are true with the substituted values (meaning the left-hand sides equal the right-hand sides), then the solution is verified. A successful verification step is a seal of approval for your algebraic solution. It is a critical step to ensure that no errors were made during the process. Skipping this step could lead to incorrect conclusions, particularly on homework or exams.