Problem 23 - Intermediate Algebra

Question

Solve each system. $$ \left\\{\begin{array}{l} 3 x+3 z=6-4 y \\ 7 x-5 z=46+2 y \\ 4 x=31-z \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 7\), \(y = -6\), \(z = 3\).

Step 1: Simplify the Equations

First, let's simplify the given equations. Each equation can be rewritten as follows:1. Equation 1: \(3x + 3z = 6 - 4y\) simplifies to \(x + z = 2 - \frac{4}{3}y\). 2. Equation 2: \(7x - 5z = 46 + 2y\) remains as is. 3. Equation 3: \(4x = 31 - z\) becomes \(z = 31 - 4x\). Now our system of equations is:\[\begin{align*}x + z &= 2 - \frac{4}{3}y, \7x - 5z &= 46 + 2y, \z &= 31 - 4x.\end{align*}\]

Step 2: Substitute Expression for z

Use the expression for \(z\) from Equation 3 and substitute it into Equations 1 and 2:- For Equation 1: \(x + (31 - 4x) = 2 - \frac{4}{3}y\) simplifies to \(-3x + 31 = 2 - \frac{4}{3}y\).- For Equation 2: \(7x - 5(31 - 4x) = 46 + 2y\) simplifies to \(27x - 155 = 46 + 2y\).This gives us two new equations in terms of \(x\) and \(y\):\[\begin{align*}3x - \frac{4}{3}y &= 29,\27x - 2y &= 201.\end{align*}\]

Step 3: Solve the System for x and y

Solve the simplified system:1. Multiply the first equation by 3 to clear the fraction: \(9x - 4y = 87\).2. Now, solve the system of equations: \[\begin{align*} 9x - 4y &= 87, \ 27x - 2y &= 201. \end{align*}\]3. To solve, multiply the first equation by 2 and the second by 4: \[\begin{align*} 18x - 8y &= 174, \ 108x - 8y &= 804. \end{align*}\]4. Subtract the first equation from the second: \(108x - 8y - (18x - 8y) = 804 - 174\) simplifies to \(90x = 630\) or \(x = 7\).

Step 4: Solve for y

With \(x = 7\), substitute back to find \(y\):1. Use \(9x - 4y = 87\): \[9(7) - 4y = 87\] simplifies to \[63 - 4y = 87\] 2. Solve for \(y\): \[-4y = 24\] \[y = -6\].

Step 5: Find z using x value

Now substitute \(x = 7\) into the expression for \(z\), \(z = 31 - 4x\):1. Substitute: \[z = 31 - 4(7)\] simplifies to \[z = 31 - 28\] 2. So, \(z = 3\).

Step 6: Verify the Solution

Finally, we verify our solution by substituting \(x = 7\), \(y = -6\), and \(z = 3\) back into the original equations to ensure they are true:1. For Equation 1: \[3(7) + 3(3) = 6 - 4(-6)\] simplifies to \[21 + 9 = 6 + 24\], which is correct.2. For Equation 2: \[7(7) - 5(3) = 46 + 2(-6)\] simplifies to \[49 - 15 = 46 - 12\], which is correct.3. For Equation 3: \[4(7) = 31 - 3\] turns into \[28 = 28\], which is also true.All equations are satisfied.

Key Concepts

Linear Equations

Linear equations are foundational tools in mathematics. They represent relationships where variables are proportional to each other. In simple terms, a linear equation can typically be written in the form of \(ax + by + cz = d\). Each term involves a constant multiplied by a variable. These equations graph as straight lines.

In a system of linear equations, like the one in our exercise, each equation represents a plane in three-dimensional space. Solving the system means finding the point(s) where all the planes intersect. Understanding linear equations helps in various fields, from physics to economics, because they elegantly model relationships that change at a constant rate.

Linear systems are important because they allow predictions. For example, if you know two variables of an equation, you can solve for the third one. This kind of predictability is valuable in planning and decision-making.

Substitution Method

The substitution method is a strategic approach to solving systems of equations. It involves solving one of the equations for one variable and then substituting this expression into another equation. This reduces the number of variables, making it easier to solve.

In our exercise, once we isolated \(z\) as \(z = 31 - 4x\), we injected this expression into the other equations. This clever move reduced the three-variable system to a two-variable problem involving just \(x\) and \(y\).

• The main advantage of substitution is its clarity. By focusing on simplifying equations step-by-step, you minimize errors and make complex systems digestible.
• Always re-check the substituted value's validity by back-substituting once the solution is found.

It’s important to choose which variable to substitute based on which one isolates most easily or reduces the complexity efficiently. Mastering this method makes tackling large systems more approachable.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to simplify and solve them, using operations like addition, subtraction, multiplication, and division. It addresses two primary goals: clearance of fractions and isolation of variables.

In our exercise, we performed algebraic manipulation by multiplying equations to clear fractions, like transforming \(3x - \frac{4}{3}y = 29\) into \(9x - 4y = 87\). This step made the equation more manageable. Furthermore, subtracting one equation from another helped to isolate \(x\) by eliminating \(y\).

Effective algebraic manipulation relies on:

• Careful calculation: Ensure that arithmetic operations are performed accurately.
• Logical order: Plan a sequence of operations that gradually simplifies the system.

This skill is crucial across all math problems; it transforms complex equations into solvable ones, laying the groundwork for deeper analysis and understanding.

Solution Verification

Solution verification is the final checkpoint in solving mathematical equations. It involves plugging the found solutions back into the original equations to ensure they meet all the conditions of the problem. Only when all conditions are satisfied can the solution be confirmed as correct.

In our exercise, we used the calculated values of \(x = 7\), \(y = -6\), and \(z = 3\) and substituted them back into the original equations. Each equation was checked individually:

• For Equation 1: We confirmed \(21 + 9 = 30\) matched the equation's right side.
• For Equation 2: Verification showed \(34 = 34\), both sides of the equation were equal.
• For Equation 3: Simplification resulted in both sides equaling 28.

This process might seem repetitive, but it prevents errors and is a vital step in ensuring the reliability and integrity of solutions.