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Chapter 2: Problem 5

Slope of the line \(2 x-y=3\) is (a) \(-2\) (b) \(-1 / 2\) (c) \(1 / 2\) (d) 2

Short Answer

Expert verified
The slope of the line \(2x - y = 3\) is \(2\), so, option (d) is the correct one.

Step by step solution

01

Convert Equation to Slope-Intercept Form

Take the given equation \(2x - y = 3\) and subtract \(2x\) from both sides to get \(-y = -2x + 3\). Next, multiply every term by \(-1\), which will give us the equation in slope-intercept form: \(y = 2x - 3\).
02

Identify the Slope

Once the equation is in the form \(y = mx + b\), the coefficient of \(x\) (the part before '\(x\)') represents the slope. Looking at our converted equation \(y = 2x - 3\), we can see that the slope \(m\) is \(2\).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is an essential concept when dealing with straight lines in mathematics. It serves as a simple structure that represents a linear equation with the formula: \[ y = mx + b \]
  • "y" is the dependent variable, representing the output that changes depending on the value of "x".
  • "m" is the slope of the line, which indicates the steepness or gradient. A positive slope means the line ascends as it moves from left to right, while a negative slope means it descends.
  • "x" is the independent variable, which is the input or the value that you can choose.
  • "b" is the y-intercept, the point where the line crosses the y-axis. This is where the x value equals zero.
This form is particularly useful because it clearly shows the slope and y-intercept values, allowing one to quickly identify both qualities of a line. When practicing, remember that converting a linear equation to this form can simplify the process of finding important graphing elements like slope and intercept.
Linear Equations
Linear equations are equations between two variables that give a straight line when plotted on a graph. These equations are commonly written in various forms, one of which is the slope-intercept form. They are crucial in analyzing relationships in mathematics due to their simplicity and straight-line graphs.Characteristics of linear equations include:
  • They have at most one degree as they do not include exponents greater than one. Each variable is of the first power.
  • They graphically represent straight lines, meaning for every point, the ratio of change in "y" to the change in "x" remains constant.
  • They are often used in real-world problem-solving, reflecting constant rates of change and proportional relationships.
In the equation \(2x - y = 3\), converting to slope-intercept form reveals the characteristics of a linear equation, allowing easy calculation and graphing.
Coefficient of x
In the slope-intercept form, the coefficient of "x" is crucial because it directly indicates the slope of the line. This slope tells us how steep the line is and in which direction it moves.Key points about the coefficient of "x":
  • In \(y = mx + b\), the "m" value, or the coefficient, represents how much "y" changes for a unit change in "x".
  • A positive "m" means the line climbs upward, while a negative "m" makes it fall downward.
  • In our reformulated equation \(y = 2x - 3\), the coefficient of "x" is 2, which means for every 1 unit you move to the right (increase in x), y increases by 2 units.
  • Understanding this helps in predicting how the line will behave in real-world applications like trend analysis and forecasting.
Therefore, identifying the coefficient of "x" is an efficient way to grasp the behavior and direction of the line in any linear equation.

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Most popular questions from this chapter

For the curve \(y=x^{3}+1\), the values of \(y\) corresponding to values of \(x\) as \(-2\) and 2 are (a) \(7,-9\) (b) \(-3,6\) (c) \(-7,9\) (d) \(6,-3\)

Derivative of \(x+\frac{1}{x}\) w.r.t. \(x\) is (a) \(1-\frac{1}{x^{2}}\) (b) \(1+\frac{1}{x^{2}}\) (c) \(-1+\frac{1}{x^{2}}\) (d) \(-1-\frac{1}{x^{2}}\)

The value of \(\frac{8 !}{6 !}\) is (a) 224 (b) 28 (c) 112 (d) 56

Expression for \(\sin C-\sin D\) is (a) \(2 \cos \frac{C-D}{2} \sin \frac{C+D}{2}\) (b) \(2 \cos \frac{C+D}{2} \sin \frac{C-D}{2}\) (c) \(2 \sin \frac{C+D}{2} \sin \frac{C-D}{2}\) (d) \(2 \cos \frac{C+D}{2} \cos \frac{C-D}{2}\)

The number of ways 5 English books and 4 Hindi books can be placed on a shelf so that the books on the same language always remain together is (a) \(5 ! 4 ! 3 !\) (b) \(5 ! 4 ! 2 !\) (c) \(4 ! 3 ! 2 !\) (d) \(5 ! 3 ! 2 !\)
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