Problem 1 - Precalculus: Mathematics for Calculus

Question

These exercises refer to the following system: $$\left\\{\begin{array}{rr}x-y+z= & 2 \\\\-x+2 y+z= & -3 \\\3 x+y-2 z= & 2\end{array}\right.$$ If we add 2 times the first equation to the second equation, the second equation becomes ___________ \(=\) ______.

Step-by-Step Solution

Answer

The second equation becomes \(x + 3z = 1\).

Step 1: Identify the First Equation

The first equation in the system is given as \(x - y + z = 2\). We will use this equation to manipulate the second equation.

Step 2: Multiply the First Equation by 2

To add 2 times the first equation to the second equation, first multiply every term of the first equation by 2. The first equation \(x - y + z = 2\) becomes:\[2(x - y + z) = 2 \times 2\] which simplifies to \(2x - 2y + 2z = 4\).

Step 3: Identify the Second Equation

The second equation in the system is \(-x + 2y + z = -3\). We need to add the transformed first equation from Step 2 to this equation.

Step 4: Add the Equations

Now we add the equations from Step 2 \(2x - 2y + 2z = 4\) and the original second equation \(-x + 2y + z = -3\). Perform the addition term-by-term for each side of the equation: \[(-x + 2y + z) + (2x - 2y + 2z) = -3 + 4\].

Step 5: Simplify the Equation

Combine like terms from the addition: - Combine \(-x\) with \(2x\) to get \(x\).- Combine \(2y\) with \(-2y\) to get \(0\).- Combine \(z\) with \(2z\) to get \(3z\).Thus, the equation simplifies to \(x + 3z = 1\).

Key Concepts

Equation Manipulation

Equation manipulation is a crucial skill when working with linear systems. It involves performing operations on one or more equations to make them easier to work with. This could include adding, subtracting, multiplying, or dividing terms to form equivalent equations.
In our example, the manipulation started with the first equation: \(x - y + z = 2\). To prepare this equation for addition with another, we multiplied every term by 2, resulting in \(2x - 2y + 2z = 4\).
This step of multiplying the entire equation helps us align the equations better, which is especially useful in eliminating variables later on. Moreover, this process of multiplying ensures the balance of the equation is still maintained, keeping the equivalence intact.

Addition of Equations

Addition of equations is a method used to simplify a system of equations by eliminating one of the variables. The main goal is to make the system easier to solve step-by-step.
For our system, we took the equation obtained after multiplying the first equation \(2x - 2y + 2z = 4\) and added it to the second equation \(-x + 2y + z = -3\).
- Add the corresponding coefficients of the variables in both equations.
- For instance, combining \(-x\) and \(2x\) gives \(x\).
- Combining \(2y\) and \(-2y\) results in \(0\), effectively eliminating the \(y\)-variable.
- Adding \(z\) to \(2z\) gives \(3z\).
On the right side, you add the constants: \(-3 + 4 = 1\).
These steps lead to a new, simplified equation: \(x + 3z = 1\). Through this process, variables are gradually eliminated from the equations—this is especially helpful to solve the system efficiently.

Secondary equations simplification

The simplification of secondary equations is an important step aimed at reducing the complexity of a system of equations. In a simplified system, equations have fewer variables, making them easier to solve.
By successfully adding and manipulating the equations, we transformed parts of the original system into a simpler equation: \(x + 3z = 1\). Having fewer variables in a single equation significantly eases solving for the others.
When you simplify secondary equations:
- Focus on reducing the number of variables they contain.
- Strive to express as many variables as possible in terms of a single variable.
- Utilize any opportunity to cancel variables out.
These strategies are essential to break down complex linear systems into more manageable forms. The eventual goal should be to isolate each variable individually across different simplified equations or to express them in relation to one another, streamlining the path to the final solution.