Problem 46 - Algebra A Combined Function

Question

Multiply both sides of equation (1) by -3 , and add the resulting equation to equation (2). $$ \begin{aligned} 2 x-3 y+2 z &=5 \\ x-9 y+z &=-1 \end{aligned} $$

Step-by-Step Solution

Answer

The final simplified equation is \(x + z = \frac{16}{5}\).

Step 1: Multiply Equation (1) by -3

Start with equation (1): \(2x - 3y + 2z = 5\). Multiply every term by -3 to get the new equation: \(-6x + 9y - 6z = -15\).

Step 2: Write Down Equation (2)

The second equation we have is \(x - 9y + z = -1\). We will be adding this to our new equation from Step 1.

Step 3: Add Resulting Equation to Equation (2)

Add the equation from Step 1, which is \(-6x + 9y - 6z = -15\), to equation (2), \(x - 9y + z = -1\). This involves adding the corresponding coefficients of \(x\), \(y\), \(z\) and the constant terms.

Step 4: Perform Addition

Perform the addition: \(-6x + x = -5x\), \(9y - 9y = 0\), \(-6z + z = -5z\), and \(-15 - 1 = -16\). So, the resulting equation is \(-5x - 5z = -16\).

Step 5: Simplify the Result

The equation \(-5x - 5z = -16\) can be simplified by dividing all terms by -5. This gives us: \(x + z = \frac{16}{5}\).

Key Concepts

Linear Equations

Linear equations are fundamental in algebra, where they represent relationships between variables through a combination of constants and variables. They take the form of a line when graphed. In equation terms, a linear equation will generally appear as y = mx + b where m represents the slope and b is the y-intercept. These equations are used to model real-world phenomena by describing proportional relationships and predictable patterns.
The distinctive feature of linear equations is that all variables are raised to the power of one, and they yield a straight line when plotted on a graph.

• For example, the equation 2x - 3y + 2z = 5 exemplifies a linear equation with three variables: x, y, and z.
• Linear equations facilitate the description of simple relationships, such as calculating earnings at a constant hourly rate.

Understanding how to manipulate and solve linear equations is crucial for solving more complex algebraic problems.

Systems of Equations

A system of equations involves multiple equations that are dealt with together, aiming to find a set of variables that satisfy all the equations in the system. These systems can hold two or more equations, often involving more than one variable. A common method for solving systems is to substitute or eliminate variables to reduce it to a single variable equation.
For instance, consider the following simple system of equations:

• 2x - 3y + 2z = 5
• x - 9y + z = -1

In working with such systems, each equation represents a separate plane or line within a multi-dimensional space.

• Using techniques such as elimination, as seen in our solution, can reduce the complexity of the system by removing variables until isolated solutions emerge.
• This approach is especially useful for determining points of intersection, which are crucial in fields like engineering and physics.

Recognizing consistent, inconsistent, and dependent systems is an essential step in effective problem solving.

Solving Equations

Solving equations involves determining the values of variables that satisfy an equation. There are several strategies depending on the type of equation, including isolating variables, substitution, and elimination. In our example, we focused on manipulating multiple equations to simplify them.
To solve the equation resulting from our steps:

• Start by isolating the variables by adding or subtracting terms from both sides to eliminate undesired terms.
• In our case, simplifying -5x - 5z = -16 to x + z = \(\frac{16}{5}\) illustrates dividing all terms by a common factor, -5.
• Finally, check if the solutions satisfy the original equation by substituting them back in.

Solving equations accurately involves not only calculation but also understanding the behavior of variables within their specific context.

Multiplication in Equations

Multiplication plays a crucial role in transforming and solving equations. By multiplying or dividing every term in an equation by the same non-zero number, you can make the equations simpler or clearer. This is particularly useful for eliminating fractions or aligning equations in a system for addition or subtraction.
In the given problem, multiplying equation (1) by -3 is crucial to effectively utilizing the elimination method. Here's why it works:

• Multiplying 2x - 3y + 2z = 5 by -3 results in -6x + 9y - 6z = -15.
• This step aligns certain terms of the two equations, enabling straightforward elimination of y when added to x - 9y + z = -1.
• Through this method, it simplifies the problem to an equation with fewer variables, making it easier to isolate a solution.

Remember that any operation performed on an equation must maintain equality, preserving the balance of both sides.