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Chapter 1: Problem 11

$$\frac{3}{4} x-\frac{1}{2} y=12$$ $$k x-2 y=22$$ If the system of linear equations above has no solution, and \(k\) is a constant, what is the value of \(k ?\) $$\begin{array}{l}{\text { (A) }-\frac{4}{3}} \\ {\text { (B) }-\frac{3}{4}} \\\ {\text { (C) } 3} \\ {\text { (D) } 4}\end{array}$$

Short Answer

Expert verified
k = 3

Step by step solution

01

Understand the System of Linear Equations

Given two linear equations: 1) \( \frac{3}{4}x - \frac{1}{2}y = 12 \) 2) \( kx - 2y = 22 \). If the system of these equations has no solution, it means the lines are parallel. For the lines to be parallel, their slopes must be equal.
02

Standardize Equation 1

Rewrite the first equation in the form \(Ax + By = C\) to clearly identify the coefficients of x and y. Multiply the entire equation by 4 to eliminate the fraction: \( 4 \cdot ( \frac{3}{4}x - \frac{1}{2}y ) = 4 \cdot 12 \). This simplifies to: \( 3x - 2y = 48 \).
03

Compare Slopes

The general form of a linear equation is \(Ax + By = C\). For the first equation, \( 3x - 2y = 48 \), the coefficient of x is 3 and the coefficient of y is -2. Therefore, the slope is \( -\frac{A}{B} = -\frac{3}{-2} = \frac{3}{2} \). For the second equation, \( kx - 2y = 22 \), the coefficient of x is k and the coefficient of y is -2. The slope is then \( -\frac{k}{-2} = \frac{k}{2} \).
04

Set Slopes Equal and Solve for k

Since the lines are parallel, their slopes must be equal: \( \frac{3}{2} = \frac{k}{2} \). Solve for k by multiplying both sides by 2: \( 3 = k \).

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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 with the same set of variables. The goal is to find the values of these variables that satisfy all the equations simultaneously.
In the given problem, we have a system of two linear equations involving variables x and y. These equations can have one unique solution (where the lines intersect at one point), no solution (where the lines are parallel), or infinitely many solutions (where the lines coincide).
Understanding this concept is crucial for solving problems related to linear equations, especially in determining whether the system has a solution and what kind of solution it has.
Parallel Lines
Parallel lines are lines in a plane that never meet. They have the same slope but different y-intercepts.
In the context of the given exercise, the system of equations has no solution because the lines are parallel. This means their slopes must be equal.
For the lines to be parallel, their slope values derived from the equations must be identical. By comparing the slopes of the given equations, you can determine when this equality holds true. Identifying parallel lines is crucial in recognizing a system of equations that has no solution.
Solving for Variables
To find the value of the variable k in the given system of equations, follow these steps:
1. **Rewrite the equations in a comparable form**: We need both equations in a format where the coefficients can be easily compared.
2. **Identify the contributions**: For each equation, determine the coefficient of x and y.
3. **Equate the slopes**: Since the lines are parallel, set their slopes equal to each other and solve for the unknown variable.
This step-by-step approach ensures that you accurately solve for the variable by adhering to algebraic principles.
Slope-Intercept Form
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
To analyze the equations given in the problem, we standardize them to the form Ax + By = C, which helps in identifying the coefficients of x and y.
1. **Simplify the equation**: Multiply by a common factor to eliminate fractions.
2. **Calculate slope**: The slope of the equation Ax + By = C is given by -A/B.
3. **Compare slopes**: Set the slopes of both equations equal to find the unknown coefficient.
This process is essential for determining the slopes of line equations and solving problems involving parallel lines.

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Most popular questions from this chapter

Oceans, seas, and bays represent about 96.5\(\%\) of Earth's water, including the water found in our atmosphere. If the volume of the water contained in oceans, seas, and bays is about \(321,000,000\) cubic miles, which of the following best represents the approximate volume, in cubic miles, of all the world's water? $$ \begin{array}{l}{\text { (A) } 308,160,000} \\ {\text { (B) } 309,765,000} \\\ {\text { (C) } 332,642,000} \\ {\text { (D) } 334,375,000}\end{array} $$

If \(y=a x^{2}+b x+c\) passes through the points \((-3,10)\) \((0,1),\) and \((2,15),\) what is the value of \(a+b+c ?\)

Which sentence would most effectively establish the main idea of the paragraph? (A) NO CHANGE (B) In addition to finding books for students, Harris is expected to meet their digital needs. (C) Librarians still perform many traditional tasks such as putting great literature in the hands of their students. (D) In the future, many school libraries are unlikely to have books on the shelves because students prefer electronic media.

If the slope of a line is \(-\frac{7}{4}\) and a point on the line is \((4,7),\) what is the \(y\) -intercept of the line?

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