Select All Statements That Are True About The Linear Equation. Y = 1 2 X − 2 Y=\frac{1}{2} X-2 Y = 2 1 ​ X − 2 A. The Graph Of The Equation Is A Single Point Representing One Solution To The Equation.B. The Point ( − 2 , − 1 (-2,-1 ( − 2 , − 1 ] Is On The Graph Of The Equation.C. The

Introduction

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on the linear equation $y=\frac{1}{2} x-2$ and analyze the given statements to determine which ones are true.

Statement A: The graph of the equation is a single point representing one solution to the equation.

To determine the validity of this statement, we need to understand the concept of a linear equation's graph. The graph of a linear equation is a straight line that passes through the points that satisfy the equation. In the case of the equation $y=\frac{1}{2} x-2$, we can rewrite it in the slope-intercept form, which is $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept.

The given equation can be rewritten as $y=\frac{1}{2} x-2$, where the slope is $\frac{1}{2}$ and the y-intercept is $-2$. This means that the graph of the equation is a straight line with a slope of $\frac{1}{2}$ and a y-intercept of $-2$.

Since the graph of the equation is a straight line, it cannot be a single point representing one solution to the equation. Therefore, Statement A is false.

Statement B: The point $(-2,-1)$ is on the graph of the equation.

To determine the validity of this statement, we need to substitute the x and y values of the point $(-2,-1)$ into the equation $y=\frac{1}{2} x-2$ and check if the equation holds true.

Substituting $x=-2$ and $y=-1$ into the equation, we get:

$-1 = \frac{1}{2} (-2) - 2$

Simplifying the equation, we get:

$-1 = -1 - 2$

$-1 = -3$

Since the equation does not hold true, the point $(-2,-1)$ is not on the graph of the equation. Therefore, Statement B is false.

Statement C: The equation has infinitely many solutions.

To determine the validity of this statement, we need to understand the concept of a linear equation's solutions. A linear equation has infinitely many solutions if it is an identity, meaning that it is true for all values of the variable(s).

However, the given equation $y=\frac{1}{2} x-2$ is not an identity. It is a linear equation with a specific slope and y-intercept. This means that it has a unique solution for each value of x.

Therefore, Statement C is false.

Conclusion

In conclusion, we have analyzed the given statements about the linear equation $y=\frac{1}{2} x-2$ and determined that Statement A and Statement B are false, while Statement C is false. The graph of the equation is a straight line with a slope of $\frac{1}{2}$ and a y-intercept of $-2$, and it has infinitely many solutions is not true.

Understanding Linear Equations: Key Concepts

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.

Graph of a Linear Equation

The graph of a linear equation is a straight line that passes through the points that satisfy the equation.

Solutions of a Linear Equation

A linear equation has infinitely many solutions if it is an identity, meaning that it is true for all values of the variable(s).

Linear Equation with a Specific Slope and Y-Intercept

A linear equation with a specific slope and y-intercept has a unique solution for each value of x.

Conclusion

In conclusion, we have analyzed the key concepts of linear equations and determined that a linear equation with a specific slope and y-intercept has a unique solution for each value of x.

Real-World Applications of Linear Equations

Physics

Linear equations are used to describe the motion of objects in physics. For example, the equation $s=vt$ describes the distance traveled by an object as a function of time, where $s$ is the distance, $v$ is the velocity, and $t$ is the time.

Engineering

Linear equations are used to design and optimize systems in engineering. For example, the equation $y=mx+b$ is used to design a linear amplifier, where $m$ is the gain and $b$ is the bias.

Economics

Linear equations are used to model economic systems. For example, the equation $y=mx+b$ is used to model the demand for a product, where $m$ is the price elasticity and $b$ is the intercept.

Conclusion

In conclusion, we have analyzed the real-world applications of linear equations and determined that they are used in various fields, including physics, engineering, and economics.

Conclusion

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Introduction

Linear equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. In this article, we will provide a comprehensive Q&A guide to help you understand linear equations better.

Q1: What is a linear equation?

A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form of $ax + by = c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.

Q2: What is the slope-intercept form of a linear equation?

The slope-intercept form of a linear equation is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q3: What is the slope of a linear equation?

The slope of a linear equation is a measure of how steep the line is. It is calculated as the ratio of the change in y to the change in x.

Q4: What is the y-intercept of a linear equation?

The y-intercept of a linear equation is the point where the line intersects the y-axis. It is the value of y when x is equal to 0.

Q5: How do I graph a linear equation?

To graph a linear equation, you can use the slope-intercept form and plot the y-intercept on the y-axis. Then, use the slope to determine the direction and steepness of the line.

Q6: What is the equation of a line that passes through the points (2,3) and (4,5)?

To find the equation of a line that passes through two points, you can use the slope formula:

m = (y2 - y1) / (x2 - x1)

Then, use the point-slope form of a linear equation:

y - y1 = m(x - x1)

Q7: How do I find the equation of a line that passes through a given point and has a given slope?

To find the equation of a line that passes through a given point and has a given slope, you can use the point-slope form of a linear equation:

y - y1 = m(x - x1)

Q8: What is the equation of a line that passes through the point (1,2) and has a slope of 3?

Using the point-slope form of a linear equation, we get:

y - 2 = 3(x - 1)

Simplifying the equation, we get:

y = 3x + 1

Q9: How do I find the equation of a line that passes through two points and has a given slope?

To find the equation of a line that passes through two points and has a given slope, you can use the slope formula:

m = (y2 - y1) / (x2 - x1)

Then, use the point-slope form of a linear equation:

y - y1 = m(x - x1)

Q10: What is the equation of a line that passes through the points (2,3) and (4,5) and has a slope of 2?

Using the slope formula, we get:

m = (5 - 3) / (4 - 2) m = 2 / 2 m = 1

Using the point-slope form of a linear equation, we get:

y - 3 = 1(x - 2)

Simplifying the equation, we get:

y = x + 1

Conclusion

In conclusion, we have provided a comprehensive Q&A guide to help you understand linear equations better. We have covered topics such as the definition of a linear equation, the slope-intercept form, the slope and y-intercept, graphing a linear equation, finding the equation of a line that passes through two points, and finding the equation of a line that passes through a given point and has a given slope.

Real-World Applications of Linear Equations

Physics

Linear equations are used to describe the motion of objects in physics. For example, the equation $s=vt$ describes the distance traveled by an object as a function of time, where $s$ is the distance, $v$ is the velocity, and $t$ is the time.

Engineering

Linear equations are used to design and optimize systems in engineering. For example, the equation $y=mx+b$ is used to design a linear amplifier, where $m$ is the gain and $b$ is the bias.

Economics

Linear equations are used to model economic systems. For example, the equation $y=mx+b$ is used to model the demand for a product, where $m$ is the price elasticity and $b$ is the intercept.

Conclusion

In conclusion, we have analyzed the real-world applications of linear equations and determined that they are used in various fields, including physics, engineering, and economics.

Practice Problems

Problem 1

Find the equation of a line that passes through the points (2,3) and (4,5).

Problem 2

Find the equation of a line that passes through the point (1,2) and has a slope of 3.

Problem 3

Find the equation of a line that passes through the points (2,3) and (4,5) and has a slope of 2.

Problem 4

Graph the equation $y=2x+1$.

Problem 5

Find the equation of a line that passes through the point (0,1) and has a slope of 2.

Answer Key

Problem 1

y = 2x + 1

Problem 2

y = 3x - 1

Problem 3

y = x + 1

Problem 4

Graph the equation y = 2x + 1.

Problem 5

y = 2x + 1

Conclusion

In conclusion, we have provided a comprehensive Q&A guide to help you understand linear equations better. We have covered topics such as the definition of a linear equation, the slope-intercept form, the slope and y-intercept, graphing a linear equation, finding the equation of a line that passes through two points, and finding the equation of a line that passes through a given point and has a given slope. We have also provided practice problems and an answer key to help you practice and reinforce your understanding of linear equations.