1. If A And B Are Two Real Numbers That Satisfy The Condition 21a = 14b, Then What Is The Value Of 3a? 2. If X Is A Rational Number, Then What Is The Solution To The Equation √2x = 2? 3. What Is The Solution To The System Of Equations {x+y=2, X-y=0, Where X And Y Are Real Numbers? 4. What Is The Ratio Of The Segments MN And PQ?
In this section, we delve into the fascinating world of real numbers and explore the relationship between two such numbers, denoted as a and b. The foundation of our analysis lies in the given condition: 21a = 14b. This equation establishes a direct proportionality between a and b, implying that their values are intertwined in a specific manner. To unravel this relationship further, our primary objective is to determine the value of 3a. This requires a careful manipulation of the given equation and a keen understanding of algebraic principles.
To begin our exploration, we can simplify the given equation by dividing both sides by a common factor. In this case, both 21 and 14 are divisible by 7. Dividing both sides of the equation 21a = 14b by 7, we obtain a simplified equation: 3a = 2b. This simplified equation reveals a more direct relationship between a and b. It states that three times the value of a is equal to two times the value of b. This insight is crucial in our quest to determine the value of 3a.
Now, let's analyze the simplified equation 3a = 2b more closely. Our objective is to find the value of 3a. From the equation, it is evident that 3a is equal to 2b. Therefore, to find the value of 3a, we need to determine the value of 2b. However, without additional information or constraints, we cannot determine a specific numerical value for 2b. The value of 2b depends on the value of b, which can be any real number.
To illustrate this point, let's consider a few examples. If b is equal to 1, then 2b is equal to 2, and therefore 3a is equal to 2. If b is equal to 2, then 2b is equal to 4, and therefore 3a is equal to 4. If b is equal to 3, then 2b is equal to 6, and therefore 3a is equal to 6. As we can see, the value of 3a varies depending on the value of b.
In conclusion, given the condition 21a = 14b, we have established that 3a = 2b. However, without additional information about the value of b, we cannot determine a specific numerical value for 3a. The value of 3a is dependent on the value of b, which can be any real number. This exploration highlights the importance of understanding the relationships between variables in mathematical equations and the limitations of solving for specific values without sufficient information.
In this section, we will focus on solving equations involving rational numbers and square roots. The specific equation we will be examining is √2x = 2, where x is a rational number (x ∈ Q). Our goal is to find the solution to this equation, which means determining the value of x that satisfies the equation. This involves understanding the properties of square roots and rational numbers, as well as applying algebraic techniques to isolate the variable x.
To begin, let's analyze the given equation: √2x = 2. This equation states that the square root of the product of 2 and x is equal to 2. To solve for x, we need to eliminate the square root. A common technique for eliminating square roots is to square both sides of the equation. Squaring both sides of the equation √2x = 2, we obtain a new equation: (√2x)² = 2². This simplifies to 2x = 4.
Now, we have a simpler equation: 2x = 4. To isolate x, we need to divide both sides of the equation by 2. Dividing both sides of the equation 2x = 4 by 2, we obtain: x = 4/2. This simplifies to x = 2. Therefore, the solution to the equation √2x = 2 is x = 2.
However, there is a crucial aspect to consider: the condition that x must be a rational number (x ∈ Q). A rational number is a number that can be expressed as a fraction p/q, where p and q are integers and q is not equal to 0. In our case, the solution we found, x = 2, is indeed a rational number because it can be expressed as the fraction 2/1. Therefore, the solution x = 2 satisfies both the equation and the condition that x is a rational number.
In conclusion, the solution to the equation √2x = 2, where x ∈ Q, is x = 2. This solution was obtained by squaring both sides of the equation to eliminate the square root, simplifying the resulting equation, and then isolating the variable x. We also verified that the solution satisfies the condition that x is a rational number. This problem demonstrates the importance of understanding the properties of square roots and rational numbers when solving mathematical equations.
In this section, we will focus on solving systems of linear equations. A system of linear equations is a set of two or more linear equations involving the same variables. The solution to a system of linear equations is the set of values for the variables that satisfy all equations in the system simultaneously. In this case, we are given the following system of equations:
{ x + y = 2,
x - y = 0,
x, y ∈ R
}
This system consists of two linear equations with two variables, x and y. The condition x, y ∈ R indicates that the solutions for x and y must be real numbers. Our goal is to find the values of x and y that satisfy both equations in the system.
There are several methods for solving systems of linear equations, including substitution, elimination, and graphical methods. In this case, we will use the elimination method. The elimination method involves manipulating the equations in the system so that when they are added together, one of the variables is eliminated. This allows us to solve for the remaining variable.
To apply the elimination method to our system, we can observe that the coefficients of y in the two equations are +1 and -1. This means that if we add the two equations together, the y terms will cancel out. Adding the two equations:
(x + y) + (x - y) = 2 + 0
This simplifies to:
2x = 2
Now, we have a single equation with one variable, x. To solve for x, we divide both sides of the equation by 2:
x = 2/2
This simplifies to:
x = 1
Now that we have found the value of x, we can substitute it back into one of the original equations to solve for y. Let's substitute x = 1 into the first equation:
1 + y = 2
To solve for y, we subtract 1 from both sides of the equation:
y = 2 - 1
This simplifies to:
y = 1
Therefore, the solution to the system of equations is x = 1 and y = 1. This means that the values x = 1 and y = 1 satisfy both equations in the system simultaneously. We can verify this by substituting these values back into the original equations:
For the first equation:
1 + 1 = 2
This is true.
For the second equation:
1 - 1 = 0
This is also true.
In conclusion, the solution to the system of equations:
{ x + y = 2,
x - y = 0,
x, y ∈ R
}
is x = 1 and y = 1. This solution was obtained using the elimination method, which involved adding the two equations together to eliminate one of the variables. We then solved for the remaining variable and substituted the value back into one of the original equations to solve for the other variable. This problem demonstrates the importance of understanding the different methods for solving systems of linear equations and the process of verifying the solutions.
In this section, we will explore the concept of ratios of line segments. A line segment is a part of a line that is bounded by two distinct endpoints. The ratio of two line segments is a comparison of their lengths, expressed as a fraction. In this case, we are asked to consider the ratio of the segments MN and PQ. To understand this ratio, we need to delve into the fundamental principles of geometry and proportionality.
The ratio of two line segments, MN and PQ, is written as MN/PQ. This ratio represents the length of segment MN divided by the length of segment PQ. The ratio can be a whole number, a fraction, or an irrational number, depending on the lengths of the segments. Understanding the ratio of line segments is crucial in various geometric applications, including similarity, proportionality, and coordinate geometry.
To illustrate the concept of ratios of line segments, let's consider a few examples. Suppose the length of segment MN is 4 units, and the length of segment PQ is 2 units. Then, the ratio of MN to PQ is 4/2, which simplifies to 2. This means that segment MN is twice as long as segment PQ. Conversely, the ratio of PQ to MN is 2/4, which simplifies to 1/2. This means that segment PQ is half as long as segment MN.
Now, let's consider another example. Suppose the length of segment MN is 5 units, and the length of segment PQ is 3 units. Then, the ratio of MN to PQ is 5/3. This ratio cannot be simplified further and represents the exact proportion between the lengths of the two segments. This means that segment MN is 5/3 times as long as segment PQ. Conversely, the ratio of PQ to MN is 3/5, which means that segment PQ is 3/5 times as long as segment MN.
The ratio of line segments is a fundamental concept in geometry and is closely related to the concept of similar figures. Two figures are said to be similar if they have the same shape but different sizes. The corresponding sides of similar figures are proportional, which means that the ratios of their lengths are equal. This property is widely used in solving geometric problems involving similar triangles and other similar figures.
In coordinate geometry, the ratio of line segments is used to find the coordinates of a point that divides a line segment in a given ratio. This is known as the section formula. The section formula is a powerful tool for solving problems related to dividing line segments in specific proportions. It has numerous applications in various fields, including computer graphics, engineering, and physics.
In conclusion, the ratio of segments MN and PQ represents the comparison of their lengths, expressed as a fraction. Understanding the ratios of line segments is crucial in various geometric applications, including similarity, proportionality, and coordinate geometry. The ratio can be a whole number, a fraction, or an irrational number, depending on the lengths of the segments. This concept provides a foundation for solving a wide range of geometric problems and is an essential tool in mathematical analysis.