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Chapter 5: Problem 11

For each given value of \(x,\) determine the value of y that gives a solution to the given linear equations in two unknowns. $$x-4 y=2 ; \quad x=3, x=-0.4$$

Short Answer

Expert verified
For \(x = 3\), \(y = 0.25\). For \(x = -0.4\), \(y = -0.6\).

Step by step solution

01

Understanding the Variables

We are given the equation: \(x - 4y = 2\). This is a linear equation, and we need to determine the value of \(y\) by substituting specific values of \(x\) given in the problem.
02

Substitute and Solve for x = 3

First, substitute \(x = 3\) into the equation. We have: \[3 - 4y = 2.\] Now, solve for \(y\) by isolating it on one side of the equation. Subtract 3 from both sides to get: \[-4y = 2 - 3\] \[-4y = -1.\] Divide both sides by \(-4\) to find \(y:\) \[y = \frac{-1}{-4} = \frac{1}{4}.\]
03

Substitute and Solve for x = -0.4

Now, substitute \(x = -0.4\) into the equation. We have: \[-0.4 - 4y = 2.\] Solve for \(y\) by first adding 0.4 to both sides: \[-4y = 2 + 0.4\] \[-4y = 2.4.\] Next, divide both sides by \(-4\) to find \(y:\) \[y = \frac{2.4}{-4} = -0.6.\]

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Key Concepts

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

The Substitution Method
The substitution method is a fundamental technique in solving systems of linear equations. This method involves substituting the value of one variable into another equation, effectively reducing the number of equations. Here, you predominantly reduce a system of equations to simpler calculations, making it easier to solve.To use the substitution method, follow these simple steps:
  • Choose a variable to solve for in one of the given equations. Typically, it's easier to solve for a variable with a coefficient of 1 or -1.
  • Substitute the expression obtained for this variable into the other equation(s). This substitution transforms the system into a simpler one-variable equation.
  • Solve this equation for that variable and back-substitute to find the other variable(s).
In our example, we used substitution twice: first replacing \(x\) with a given value and solving for \(y\). By systematically replacing and solving, you can solve various systems of equations effectively.
Solving for Variables
When solving linear equations, the main goal is to isolate the variable of interest. This makes the value of the variable evident. Understanding the steps to solve for a variable can greatly improve one's problem-solving skills in algebra. Here's the step-by-step approach:
  • Begin by simplifying each side of the equation if necessary. Combine like terms or simplify fractions as needed.
  • Isolate the variable on one side of the equation. You usually do this by adding or subtracting terms from both sides and then multiplying or dividing by the variable's coefficient.
  • Check your solutions by substituting them back into the original equation to ensure both sides are equal.
For example, we isolated \(y\) by subtracting and then dividing other terms. Precision in each step is critical to make sure the solution is correct.
Mathematics Education - Building Understanding
Mathematics education revolves around developing problem-solving skills and logical reasoning through clear comprehension of concepts. It is crucial for students to master techniques like solving linear equations, as these are foundational skills in math-related fields. Concepts like the substitution method don't just build arithmetic skills, but also improve:
  • Analytical thinking: Learning to break down a problem and tackle it systematically.
  • Critical reasoning: Understanding the 'whys' and 'hows' of each step in problem-solving, making math less about rote memorization and more about genuine comprehension.
  • Confidence in problem-solving: Familiarity with processes like substitution enhances self-reliance when faced with new challenges.
By incorporating these methods into regular practice, students not only perform well in exams but also gain skills that are applicable in everyday decision-making processes.

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Most popular questions from this chapter

Evaluate the given third-order determinants. $$\left|\begin{array}{rrr} 10 & 2 & -7 \\ -2 & -3 & 6 \\ 6 & 5 & -2 \end{array}\right|$$

Show that the given systems of equations have either an unlimited number of solutions or no solution. If there is an unlimited number of solutions, find one of them. $$\begin{aligned}&3 x+y-z=-3\\\&x+y-3 z=-5\\\&-5 x-2 y+3 z=-7\end{aligned}$$

Use the determinant at the right. Answer the questions about the determinant for the changes given in each exercise. \(\left|\begin{array}{lll}4 & 2 & 1 \\\ 3 & 6 & 5 \\ 7 & 9 & 8\end{array}\right|=19\) How does the value change if the first two rows are interchanged?

Solve the given systems of equations by the method of elimination by addition or subtraction. $$\begin{aligned} &3 x-y=3\\\ &4 x=3 y+14 \end{aligned}$$

Set up appropriate systems of equations. All numbers are accurate to at least two significant digits. A person spent \(1.10 \mathrm{h}\) in a car going to an airport, \(1.95 \mathrm{h}\) flying in a jet, and \(0.520 \mathrm{h}\) in a taxi to reach the final destination. The jet's speed averaged 12.0 times that of the car, which averaged \(15.0 \mathrm{mi} / \mathrm{h}\) more than the taxi. What was the average speed of each if the trip covered \(1140 \mathrm{mi} ?\)
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