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Chapter 1: Problem 46

Solve the system of linear equations. $$\begin{aligned}&0.05 x_{1}-0.03 x_{2}=0.21\\\&0.07 x_{1}+0.02 x_{2}=0.17\end{aligned}$$

Short Answer

Expert verified
The solution to the system of linear equations is: \(x1=1.2\) and \(x2=1\)

Step by step solution

01

Arrange Appropriately

First, rearrange each equation to isolate \(x1\).\nFor the first equation: \(x1=(0.21 + 0.03x2) / 0.05\). For the second equation: \(x1=(0.17 - 0.02x2) / 0.07\)
02

Set the expressions for \(x1\) equal to each other

Since both expressions are equal to \(x1\), they can be set equal to each other to get: \((0.21 + 0.03x2) / 0.05 = (0.17 - 0.02x2) / 0.07\)
03

Cross Multiplication and Solve for \(x2\)

Cross-multiply to eliminate the denominators and simplify the equation to find the value of \(x2\). This gives us: \(x2 = (0.07*0.21 + 0.07*0.03x2) = (0.05*0.17 - 0.05*0.02x2)\).\nSolving this equation for \(x2\) we find that: \(x2 = 1\).
04

Substitute \(x2\) into one of the original equations

Substitute \(x2=1\) into the first original equation: \(0.05x1 - 0.03*1 = 0.21\). Solving this for \(x1\), you get: \(x1=0.06/0.05\).\nSo: \(x1=1.2\).

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Key Concepts

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

System of Equations
A system of equations consists of two or more equations that share the same variables. In our example, we deal with a set of linear equations involving two variables, \( x_1 \) and \( x_2 \). The goal is to find values for these variables that satisfy both equations simultaneously. Here are some key points about systems of equations:
  • Systems can have one solution, no solution, or infinitely many solutions.
  • Each equation represents a line in a two-dimensional space; a solution is where these lines intersect.
  • Various methods, such as substitution and elimination, can be used to find solutions.
Understanding systems is crucial in solving problems involving multiple interconnected variables, which is common in real-world scenarios.
Cross Multiplication
Cross-multiplication is a powerful technique used to solve equations involving fractions. It helps eliminate the fractions by transforming the equation into a simpler form. This technique is especially useful in systems of equations where variables are in fractional form. Let鈥檚 examine how it is employed:
  • Begin by identifying two fractions set equal to each other, like in the derived equation from the exercise: \( \frac{0.21 + 0.03x_2}{0.05} = \frac{0.17 - 0.02x_2}{0.07} \)
  • Next, cross-multiply by multiplying the numerator of each fraction by the denominator of the other fraction: \( 0.07(0.21 + 0.03x_2) = 0.05(0.17 - 0.02x_2) \).
  • This step eliminates the fractions, allowing you to solve a linear equation, which is generally much easier to handle.
Cross-multiplication aids in simplifying the solution process, making it more manageable and less prone to errors.
Substitution Method
The substitution method involves solving one of the equations for one variable and then substituting this expression into the other equation. This can simplify a system into a single equation that is easier to solve. Here's a breakdown of the substitution method:
  • First, isolate one variable in one of the equations, like solving for \( x_1 \) in each equation from our example.
  • Next, substitute the expression for the isolated variable into the other equation. This transforms the system into one with only a single variable.
  • Solve for this remaining variable, which is \( x_2 \) in our example, making it easier to find its value.
  • Finally, take the found value and substitute it back into any of the original equations to determine the value of the second variable, \( x_1 \) in this case.
Using substitution is beneficial as it reduces complex multivariable equations into simpler single-variable equations, paving the way for straightforward solutions.

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Most popular questions from this chapter

True or False? In Exercises 69 and \(70,\) determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) A linear system can have exactly two solutions. (b) Two systems of linear equations are equivalent when they have the same solution set. (c) A system of three linear equations in two variables is always inconsistent.

Sketch the lines represented by the system of equations. Then use Gaussian elimination to solve the system. At each step of the elimination process, sketch the corresponding lines. What do you observe about the lines? $$\begin{array}{rr}2 x-3 y= & 7 \\\\-4 x+6 y= & -14\end{array}$$

System of Linear Equations In Exercises \(39-42\), use \(\Rightarrow\) a software program or a graphing utility to solve the system of linear equations. $$ \begin{aligned} &x_{1}-2 x_{2}+5 x_{3}=30\\\ &\begin{aligned} x_{1}+4 x_{2}-7 x_{3}-2 x_{4} &=45.7 \\ 3 x_{1}-5 x_{2}+7 x_{3}+4 x_{4} &=29.9 \end{aligned} \end{aligned} $$

Find the reduced row-echelon matrix that is row-equivalent to the given matrix. $$ \left[\begin{array}{rr} 1 & 2 \\ -1 & 2 \end{array}\right] $$

Sketch the lines represented by the system of equations. Then use Gaussian elimination to solve the system. At each step of the elimination process, sketch the corresponding lines. What do you observe about the lines? $$\begin{aligned}x-4 y &=-3 \\\5 x-6 y &=13\end{aligned}$$
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