Mastering Linear Equations Identifying And Verifying Solutions

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1. Which of the following equations is linear? a) $3x^2 + 9y + z = 5$ b) $2x^2 + \frac{2}{y} + z = 3$ c) $\frac{x}{2} - \frac{2y}{3} + 7z = 81$ d) $2x^2 + y^2 + 3z = 1$ 2. Which of the following ordered pairs is a solution of the linear equation $x - 2y = 4$? a) (10,1) b) (2, -1)

In the realm of mathematics, linear equations hold a fundamental position, serving as the building blocks for more complex mathematical models and applications. Understanding what constitutes a linear equation is crucial for navigating various mathematical concepts and problem-solving scenarios. A linear equation is characterized by its variables being raised to the power of one, and there are no products or other complex functions involving the variables. Let's delve into the characteristics that define a linear equation and then analyze the given options to pinpoint the one that fits the criteria.

To effectively identify a linear equation, it's essential to grasp the defining traits that set it apart from its non-linear counterparts. At its core, a linear equation is an algebraic equation where the highest power of any variable is 1. This means that you won't encounter terms like x², y³, or any other variable raised to a power greater than 1. Additionally, linear equations do not involve variables multiplied by each other (like xy) or variables appearing in denominators (like 1/x). These restrictions ensure that the relationship between the variables is a straight line when graphed, hence the term "linear". The general form of a linear equation in multiple variables is often expressed as a₁x₁ + a₂x₂ + ... + aₙxₙ = b, where a₁, a₂, ..., aₙ are constants, x₁, x₂, ..., xₙ are variables, and b is a constant. This form highlights the direct, proportional relationship between the variables and the constant term.

Now, let's put this understanding into practice by examining the given options and determining which one adheres to the linear equation criteria. We'll dissect each equation, paying close attention to the powers of the variables and the presence of any non-linear terms. This process will not only help us identify the correct linear equation but also reinforce our understanding of the key characteristics that define them. By the end of this analysis, you'll be well-equipped to confidently identify linear equations in various mathematical contexts, paving the way for tackling more advanced concepts and problem-solving scenarios. Remember, the ability to recognize linear equations is a foundational skill in mathematics, opening doors to a wide array of applications and theoretical explorations.

Analyzing the Options

Let's analyze each of the provided equations to determine which one is linear:

a) 3x2+9y+z=53x^2 + 9y + z = 5

This equation contains the term 3x23x^2, where the variable x is raised to the power of 2. This immediately disqualifies it from being a linear equation, as linear equations only allow variables to be raised to the power of 1.

b) 2x^2 + rac{2}{y} + z = 3

Here, we have two terms that violate the linearity criteria. First, the term 2x22x^2 has x raised to the power of 2. Second, the term 2y\frac{2}{y} can be rewritten as 2y−12y^{-1}, indicating that y is raised to the power of -1. Both of these terms make the equation non-linear.

c) x2−2y3+7z=81\frac{x}{2} - \frac{2y}{3} + 7z = 81

This equation appears to fit the definition of a linear equation. We can rewrite it as 12x−23y+7z=81\frac{1}{2}x - \frac{2}{3}y + 7z = 81. All variables (x, y, and z) are raised to the power of 1, and there are no products of variables or variables in denominators. This equation adheres to the form a₁x₁ + a₂x₂ + ... + aₙxₙ = b, where the coefficients are constants and the variables are raised to the power of 1. Therefore, this is a linear equation.

d) 2x2+y2+3z=12x^2 + y^2 + 3z = 1

This equation contains the terms 2x22x^2 and y2y^2, where the variables x and y are raised to the power of 2. This violates the linearity condition, making the equation non-linear.

Therefore, the linear equation among the options is c) x2−2y3+7z=81\frac{x}{2} - \frac{2y}{3} + 7z = 81.

Once we've identified a linear equation, a common task is to determine whether a given set of values for the variables (an ordered pair or a tuple) constitutes a solution. A solution to a linear equation is a set of values that, when substituted into the equation, make the equation true. This section will guide you through the process of verifying solutions for linear equations, focusing on the equation x−2y=4x - 2y = 4 and the provided ordered pairs.

The concept of a solution to a linear equation is fundamental to understanding the relationship between the variables and the equation itself. In essence, a solution is a specific combination of values for the variables that satisfies the equation's equality. For an equation with two variables, like x−2y=4x - 2y = 4, a solution is represented as an ordered pair (x, y). To verify if an ordered pair is a solution, you simply substitute the x and y values into the equation and check if the resulting statement is true. If the left-hand side of the equation equals the right-hand side after the substitution, then the ordered pair is a solution. If not, it's not a solution. This process highlights the core principle of equations: they represent a balance, and a solution is a set of values that maintains that balance. The ability to verify solutions is crucial for solving linear equations, graphing linear equations, and understanding systems of linear equations. It provides a concrete way to check your work and ensure that you've found the correct values for the variables.

Now, let's apply this concept to the given equation and ordered pairs. We'll take each ordered pair, substitute the x and y values into the equation x−2y=4x - 2y = 4, and see if the equation holds true. This hands-on approach will solidify your understanding of solution verification and equip you with the skills to confidently tackle similar problems. Remember, the key is to carefully substitute the values and perform the arithmetic operations accurately. By the end of this exercise, you'll be proficient in verifying solutions for linear equations, a valuable skill in various mathematical contexts.

Testing the Ordered Pairs

To determine which of the given ordered pairs is a solution to the equation x−2y=4x - 2y = 4, we will substitute the x and y values from each pair into the equation and check if the equation holds true.

a) (10, 1)

Substitute x = 10 and y = 1 into the equation:

10−2(1)=410 - 2(1) = 4

10−2=410 - 2 = 4

8=48 = 4

This statement is false. Therefore, (10, 1) is not a solution to the equation.

b) (2, -1)

Substitute x = 2 and y = -1 into the equation:

2−2(−1)=42 - 2(-1) = 4

2+2=42 + 2 = 4

4=44 = 4

This statement is true. Therefore, (2, -1) is a solution to the equation.

Therefore, the ordered pair (2, -1) is a solution of the linear equation x−2y=4x - 2y = 4.

In this guide, we've explored the fundamental concepts of linear equations, including how to identify them and how to verify their solutions. We've learned that a linear equation is characterized by variables raised to the power of 1 and the absence of non-linear terms. We've also demonstrated the process of substituting ordered pairs into linear equations to determine if they satisfy the equation. These skills are essential for further studies in mathematics and its applications.