Problem 10 - Algebra and Trigonometry

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{array}{l}{3 x+2 y=0} \\ {-x-2 y=8}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \((4, -6)\).

Step 1: Choose a Method

Choose a method to solve the system of equations. In this case, we will use the method of elimination to find the solution.

Step 2: Align Equations

Align the given system of equations: 1) \(3x + 2y = 0\) 2) \(-x - 2y = 8\).

Step 3: Add the Equations

Add the two equations to eliminate the variable \(y\). Observe:\[3x + 2y - (x + 2y) = 0 + 8\]This simplifies to:\[2x = 8\].

Step 4: Solve for x

Solve the equation \(2x = 8\) for \(x\). Divide both sides by 2:\[x = 4\].

Step 5: Substitute to Find y

Substitute \(x = 4\) back into one of the original equations to find \(y\). Using the first equation:\[3(4) + 2y = 0\]This simplifies to:\[12 + 2y = 0\]Subtract 12 from both sides:\[2y = -12\]. Divide by 2:\[y = -6\].

Step 6: Write the Solution

Write the solution to the system as an ordered pair. The solution is \((4, -6)\).

Key Concepts

Method of Elimination

The method of elimination is a technique used to solve systems of linear equations. It involves adding or subtracting equations in a way that removes one of the variables, making it possible to solve for the remaining variable. This method is particularly useful when the coefficients of one of the variables are easily manipulated to cancel each other out.

For instance, in the given system:

• 3x + 2y = 0
• -x - 2y = 8

we can readily eliminate y by adding these two equations. The opposing coefficients of y (2 and -2) allow for straightforward cancelation, simplifying the solution process. After elimination, you will have a single-variable equation, which can then be solved as usual. This approach requires careful alignment of equations and accurate arithmetic to avoid mistakes.

Ultimately, elimination is a powerful strategy for efficiently solving systems when substitution or graphing might be cumbersome or less clear.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when graphed. They typically take the form ax + by = c, where a, b, and c are constants. These equations can contain one or more variables, which we aim to solve during the problem.

In the system:

• 3x + 2y = 0
• -x - 2y = 8

you are working with two linear equations that intersect at a single point. This intersection point is the solution to the system, indicating the values of x and y that satisfy both equations simultaneously.

Linear equations are foundational in algebra and have wide applications, from calculating slopes and intercepts to modeling real-world situations. They’re called ‘linear’ because they graph as straight lines. Understanding linear equations is crucial for moving on to more complex algebraic concepts.

Ordered Pair Solution

An ordered pair solution is a set of values that represents the point of intersection for a system of equations in the coordinate plane. It is usually written in the form (x, y). This structure ensures that the specific values for each variable are clearly identified.

When solving the given system of linear equations:

• 3x + 2y = 0
• -x - 2y = 8

following the method of elimination gives you the ordered pair solution of (4, -6). This indicates that x is 4 and y is -6.

Such solutions are key in many math-related fields, linking algebraic expressions to graph interpretation. They confirm the exact location on the graph where two lines meet, offering a visual and numerical representation of their relationship.