Question

Solve the system by elimination. Then use the graph to confirm your solution. Copy the graph and label each line with the appropriate equation. $$ \left\\{\begin{aligned} 3 x-2 y &=2 \\ x+2 y &=6 \end{aligned}\right. $$

Short answer

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

Step-by-step solution

Performing the Elimination

To execute the elimination process, add the two equations. If the coefficients of y in both equations are opposite, they will cancel out each other.\n\nSo, adding \(3x - 2y = 2\) and \(x + 2y = 6\) gives \(4x = 8\).

Solving for x

After obtaining \(4x = 8\), divide both sides by 4 to obtain the value of x. Solving gives \(x = 2\).

Substituting x into the second equation

Substitute \(x = 2\) into the second equation \(x + 2y = 6\) to find the value of y. This will give \(2 + 2y = 6\), and simplifying it will result in \(2y = 4\), so \(y = 2\).

Verifying the solution using the graph

Plot the two lines \(3x - 2y = 2\) and \(x + 2y = 6\) on a graph. The point of intersection should be the solution of the system (2,2) obtained from the previous steps. If the point (2,2) lies on both lines, this indicates that our solution is correct.

Key concepts

Elimination Method

The elimination method is an effective algebraic technique used to solve systems of linear equations. It involves manipulating the equations in such a way that adding or subtracting them from each other results in one of the variables being eliminated. In our exercise, we targeted the variable y for elimination. This approach requires a keen eye to identify coefficients that can be canceled out with mathematical operations.

For beginners, it's helpful to visualize the elimination process as the balancing of a scale – changes made to one side must also be made to the other to maintain equilibrium. Here's a pro tip: always aim for the smallest coefficients to cancel out first; this often simplifies calculations and prevents dealing with fractions unnecessarily.

System of Linear Equations

A system of linear equations consists of two or more equations with one or more variables. These equations represent lines in a coordinate plane, and their point of intersection reflects the solution to the system. There are three possible outcomes: a single point of intersection (one solution), overlapping lines (infinitely many solutions), or parallel lines with no intersection (no solution).

The system from our exercise has two equations with two variables (x and y), categorizing it as a two-variable linear system. Such systems can be solved using various methods including substitution, elimination (the focal method in our scenario), and graphing.

Graphical Method Verification

After finding a solution algebraically, verifying it graphically ensures its accuracy. The graphical method involves plotting the equations on a coordinate plane and identifying their point of intersection.

For our exercise, drawing lines for the equations 3x - 2y = 2 and x + 2y = 6 on a graph should yield a point where both lines cross, confirming the solution obtained via algebra. This visual representation can be particularly reassuring for visual learners, as it provides a tangible affirmation of the abstract algebraic work. Remember, if the point of intersection correlates with the solution derived algebraically, the solution is correct!

Algebraic Solution

The algebraic solution is the result obtained through systematic manipulation of equations using algebraic principles, like the elimination method. It includes solving for the unknown variables, often yielding definitive numerical values.

In this exercise, we found that x = 2 and y = 2 by first eliminating one of the variables and then substituting back to solve for the other. Completing these algebraic steps accurately is crucial; one arithmetic mistake can derail the entire solution. Therefore, it's vital to check each algebraic step carefully and to use verification methods, such as the graphical approach, to ensure the correctness of the solution.