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Chapter 5: Problem 12

Solve the system of equations. $$\left\\{\begin{aligned} 6 y+4 z &=-12 \\ 3 x+3 y &=9 \\ 2 x-3 z &=10 \end{aligned}\right.$$

Short Answer

Expert verified
The solution to the system of equations are \(x = 2, y = 1, z = -1\).

Step by step solution

01

Rearranging Terms in First Equation

Initially, divide the second equation by 3 to simplify it i.e. \(x+y=3\). Thus, you can attain \(y\).Next, from the first equation, express \(z\) in terms of \(y\) by rearranging terms: \(z = -(\frac{6y+12}{4}) = -(1.5y + 3)\). This will allow eliminating \(z\) later on.
02

Substitute \(y\) and \(z\) into Third Equation

Now, express \(y\) in terms of \(x\) from the second equation: \(y = 3 - x\). Replace \(y\) and \(z\) in the third equation as mentioned: \(2x - 3[-(1.5(3-x) + 3)] = 10\).
03

Solving for \(x\)

Calculation on the above equation: \(2x - 3[-(4.5 - 1.5x) + 3] = 10\) simplifies to \(2x - 3[-1.5x + 3] = 10\), which further simplifies to \(2x + 4.5x - 9 = 10\). Solving with respect to \(x\) yields \(x = 2\).
04

Solving for \(y\) and \(z\)

Substitute \(x = 2\) in the second equation, \(x + y = 3\). You attain the value of \(y = 1\). Replace \(y = 1\) in the first equation, where you find \(z = -1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are among the simplest types of algebraic expressions, making them a cornerstone in algebra.
Each equation represents a straight line in either a two-dimensional or multi-dimensional space.
Linear equations usually come in the form of \( ax + by + cz = d \), where \( a, b, c, \) and \( d \) are constants, and \( x, y, \) and \( z \) are variables.
  • The main feature of linear equations is that each term is either a constant or the product of a constant and a single variable.
  • They do not include exponents, products of variables, or any higher-level terms (like squared or cubed terms).
  • Since they form lines, any system involving two linear equations can be visualized as the intersection of two planes, where the solution is essentially the point(s) where these planes intersect.
Linear equations are utilized primarily to model real-world scenarios because they offer a simple yet effective representation of mathematical relationships. Thanks to their straightforward nature, systems of linear equations are quite useful in fields such as economics, physics, engineering, and countless others.
Solving Equations
Solving equations, especially in a system, involves finding values for the variables that make all the equations in the system true.
We typically solve systems of equations to find the points where lines or planes intersect in space.
In our original exercise, the system consisted of three equations and three unknowns, \( x, y, \) and \( z \).
  • Start by isolating one variable, as we've seen with expressing \( y \) in terms of \( x \), which simplifies the solving process.
  • Substitution is a powerful method where you replace one variable with an expressed term in terms of other variables.
  • After substitution, equations simplify, as shown when \( y \) and \( z \) values were replaced in the third equation, allowing the solving of \( x \).
With these strategies, solving a system of equations becomes a structured process. Take one step at a time, isolate variables, substitute, and solve step-by-step to reveal each unknown.
Algebraic Manipulation
Algebraic manipulation is a process used to change the form of an equation while maintaining equality.
This skill is crucial for successfully solving systems of equations, as it allows for isolating variables and reducing complex expressions.
  • Rearranging terms involves adding, subtracting, or factoring terms to simplify or solve parts of an equation, as done when we express \( z \) in terms of \( y \).
  • Distributing negative signs and constants is another vital step; for example, distributing the \( -(1.5y + 3) \) allowed us to eliminate \( z \).
  • Combining like terms reduces the number of terms and helps in simplifying equations further, as seen when terms involving \( x \) were combined to solve for \( x \).
With consistent practice in algebraic manipulation, solving a system of equations becomes more intuitive and less daunting. The key is to keep the equation balanced 鈥� what you do to one side must be done to the other.

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Most popular questions from this chapter

The given linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the maximum value of the objective function and where it occurs. Objective function: \(z=x+2 y\) Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+2 y & \leq 4 \\ 2 x+y & \leq 4 \end{aligned} $$

Computers The sales \(y\) (in billions of dollars) for Dell Inc. from 1996 to 2005 can be approximated by the linear model \(y=5.07 t-22.4, \quad 6 \leq t \leq 15\) where \(t\) represents the year, with \(t=6\) corresponding to 1996\. (Source: Dell Inc.) (a) The total sales during this ten-year period can be approximated by finding the area of the trapezoid represented by the following system. \(\left\\{\begin{array}{l}y \leq 5.07 t-22.4 \\ y \geq 0 \\ t \geq 5.5 \\ t \leq 15.5\end{array}\right.\) Graph this region using a graphing utility. (b) Use the formula for the area of a trapezoid to approximate the total sales.

Writing Explain the difference between the graphs of the inequality \(x \leq 4\) on the real number line and on the rectangular coordinate system.

Investments An investor has up to $$\$ 450,000$$ to invest in two types of investments. Type A investments pay \(8 \%\) annually and type \(B\) pay \(14 \%\) annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to type A investments and at least one-fourth is to be allocated to type \(\mathrm{B}\) investments. What is the optimal amount that should be invested in each type of investment? What is the optimal return?

Optimal Profit A manufacturer produces two models of elliptical cross-training exercise machines. The times for assembling, finishing, and packaging model \(\mathrm{A}\) are 3 hours, 3 hours, and \(0.8\) hour, respectively. The times for model B are 4 hours, \(2.5\) hours, and \(0.4\) hour. The total times available for assembling, finishing, and packaging are 6000 hours, 4200 hours, and 950 hours, respectively. The profits per unit are \(\$ 300\) for model \(A\) and $$\$ 375$$ for model \(B\). What is the optimal production level for each model? What is the optimal profit?
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