In the realm of algebra, equations form the bedrock of mathematical problem-solving. Among the diverse forms of equations, linear equations hold a prominent position due to their simplicity and wide applicability. This article delves into the intricacies of a specific linear equation, 5y + 16x = 4y + 10, providing a comprehensive analysis and step-by-step solution to illuminate its underlying principles.
The equation 5y + 16x = 4y + 10 may seem daunting at first glance, but with a systematic approach, it can be deciphered and solved. This equation is a linear equation, which means that it represents a straight line when graphed on a coordinate plane. Linear equations are characterized by their variables being raised to the power of 1, and they can be expressed in various forms, such as slope-intercept form, standard form, and point-slope form.
The equation 5y + 16x = 4y + 10 contains two variables, x and y, making it a two-variable linear equation. To solve for the values of x and y that satisfy this equation, we need to manipulate it algebraically. The goal is to isolate one variable on one side of the equation and express it in terms of the other variable. This process involves combining like terms, applying the distributive property, and performing other algebraic operations.
Simplifying the Equation
Simplifying the equation 5y + 16x = 4y + 10 is the initial step in unraveling its solution. This process involves rearranging the terms to group like terms together, making it easier to isolate variables and ultimately solve for their values. The primary objective is to consolidate similar terms on each side of the equation, paving the way for further algebraic manipulations.
Combining Like Terms
The equation 5y + 16x = 4y + 10 consists of several terms, each with its own variable or constant. To simplify the equation, we must identify and combine like terms. Like terms are those that share the same variable raised to the same power. In this equation, we have two terms with the variable y: 5y and 4y. We also have a term with the variable x: 16x. And finally, we have a constant term: 10.
To combine the y terms, we subtract 4y from both sides of the equation. This step ensures that the equation remains balanced, as we are performing the same operation on both sides. Subtracting 4y from both sides yields:
5y + 16x - 4y = 4y + 10 - 4y
Simplifying this equation, we get:
y + 16x = 10
Now, we have successfully combined the y terms, reducing the complexity of the equation. The equation now contains only one y term, one x term, and a constant term. This simplification brings us closer to isolating the variables and solving for their values.
Rearranging Terms
To further simplify the equation, we can rearrange the terms to group the variables on one side and the constant on the other side. This rearrangement is a standard algebraic technique that helps to isolate variables and make the equation easier to solve. — Men's Basketball Shoes Sale: Top Deals & Brands
To rearrange the terms, we subtract 16x from both sides of the equation. This step moves the x term to the right side of the equation, leaving the y term on the left side. Subtracting 16x from both sides yields:
y + 16x - 16x = 10 - 16x
Simplifying this equation, we get:
y = 10 - 16x
Now, we have successfully rearranged the terms, isolating the variable y on the left side of the equation. The equation is now in slope-intercept form, which is a standard form for linear equations. This form allows us to easily identify the slope and y-intercept of the line represented by the equation.
Isolating the Variable 'y'
Isolating the variable 'y' is a crucial step in solving the equation 5y + 16x = 4y + 10. This process involves manipulating the equation to get 'y' by itself on one side, allowing us to express 'y' in terms of 'x'. Isolating 'y' provides valuable insights into the relationship between the variables and paves the way for finding solutions to the equation.
Subtracting 4y from Both Sides
The initial step in isolating 'y' involves eliminating the '4y' term from the right side of the equation. To achieve this, we subtract '4y' from both sides of the equation. This operation maintains the balance of the equation while effectively removing the '4y' term from the right side.
Subtracting '4y' from both sides of the equation yields:
5y + 16x - 4y = 4y + 10 - 4y
Simplifying this equation, we get:
y + 16x = 10
Now, we have successfully eliminated the '4y' term from the right side of the equation, bringing us closer to isolating 'y'. The equation now contains only one 'y' term, along with the '16x' term and the constant term '10'.
Subtracting 16x from Both Sides
To further isolate 'y', we need to eliminate the '16x' term from the left side of the equation. To accomplish this, we subtract '16x' from both sides of the equation. This operation maintains the balance of the equation while effectively removing the '16x' term from the left side.
Subtracting '16x' from both sides of the equation yields:
y + 16x - 16x = 10 - 16x
Simplifying this equation, we get:
y = 10 - 16x
Now, we have successfully isolated 'y' on the left side of the equation. The equation now expresses 'y' in terms of 'x', which means that for any given value of 'x', we can calculate the corresponding value of 'y'. This isolated form of the equation provides a clear understanding of the relationship between the variables and is essential for solving the equation.
Expressing 'y' in Terms of 'x'
Expressing 'y' in terms of 'x' is a fundamental technique in algebra, particularly when dealing with linear equations. This process involves manipulating the equation to isolate 'y' on one side, allowing us to express 'y' as a function of 'x'. This form of the equation provides valuable insights into the relationship between the variables and facilitates solving for specific values of 'y' given corresponding values of 'x'.
The Equation y = 10 - 16x
After performing the algebraic manipulations described in the previous sections, we arrive at the equation:
y = 10 - 16x
This equation expresses 'y' in terms of 'x'. It states that the value of 'y' is equal to 10 minus 16 times the value of 'x'. This form of the equation is known as slope-intercept form, which is a standard form for linear equations. Slope-intercept form allows us to easily identify the slope and y-intercept of the line represented by the equation.
Understanding Slope-Intercept Form
The slope-intercept form of a linear equation is generally written as:
y = mx + b
where:
- 'y' is the dependent variable
- 'x' is the independent variable
- 'm' is the slope of the line
- 'b' is the y-intercept of the line
In the equation y = 10 - 16x, we can identify the following:
- The slope (m) is -16. This means that for every 1 unit increase in 'x', the value of 'y' decreases by 16 units.
- The y-intercept (b) is 10. This means that the line intersects the y-axis at the point (0, 10).
Applications of Expressing 'y' in Terms of 'x'
Expressing 'y' in terms of 'x' has numerous applications in mathematics, science, and engineering. Some of these applications include:
- Graphing linear equations: The slope-intercept form makes it easy to graph a linear equation. We can plot the y-intercept and then use the slope to find other points on the line.
- Solving for 'y' given 'x': If we know the value of 'x', we can easily substitute it into the equation y = 10 - 16x to find the corresponding value of 'y'.
- Modeling real-world relationships: Linear equations can be used to model a variety of real-world relationships, such as the relationship between distance and time, or the relationship between cost and quantity. Expressing 'y' in terms of 'x' allows us to understand how one variable changes in response to changes in the other variable.
Solutions to the Equation
The equation 5y + 16x = 4y + 10, after simplification and manipulation, yields y = 10 - 16x. This equation represents a linear relationship between the variables x and y. Unlike equations with a single solution, this equation has infinitely many solutions. Each solution is a pair of values (x, y) that satisfies the equation. These solutions can be visualized as points on the line represented by the equation on a coordinate plane.
Infinite Solutions
The presence of two variables in a single linear equation leads to an infinite number of solutions. For every value we assign to x, we can calculate a corresponding value for y, and vice versa. This means there's no single, unique answer for x or y; instead, there's a range of pairs that make the equation true.
Finding Solutions
To find specific solutions, we can choose a value for x and substitute it into the equation y = 10 - 16x to find the corresponding value for y. Alternatively, we can choose a value for y and solve for x. This process can be repeated indefinitely, generating an infinite number of solutions.
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Example 1: Let's choose x = 0.
Substituting x = 0 into the equation y = 10 - 16x, we get:
y = 10 - 16(0) = 10
So, one solution is (0, 10).
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Example 2: Let's choose x = 1.
Substituting x = 1 into the equation y = 10 - 16x, we get:
y = 10 - 16(1) = -6
So, another solution is (1, -6).
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Example 3: Let's choose y = 0.
Substituting y = 0 into the equation y = 10 - 16x, we get:
0 = 10 - 16x
Solving for x, we get:
16x = 10
x = 10/16 = 5/8
So, another solution is (5/8, 0).
These examples demonstrate how we can generate solutions by choosing values for one variable and solving for the other. Each pair of values (x, y) that we find in this way represents a point on the line defined by the equation y = 10 - 16x.
Graphical Representation
The solutions to the equation 5y + 16x = 4y + 10 can be graphically represented as a straight line on a coordinate plane. Each point on the line corresponds to a solution (x, y) that satisfies the equation. The line extends infinitely in both directions, visually representing the infinite number of solutions.
The slope of the line is -16, which indicates the rate at which y changes with respect to x. The y-intercept is 10, which is the point where the line crosses the y-axis. By plotting the line, we can visualize the set of all possible solutions to the equation.
Real-World Applications
Linear equations, such as 5y + 16x = 4y + 10, are not just abstract mathematical concepts; they have numerous real-world applications across various fields. These equations can be used to model and solve problems in areas like physics, engineering, economics, and computer science. Understanding the applications of linear equations enhances our ability to analyze and interpret real-world phenomena.
Modeling Relationships
Linear equations are particularly useful for modeling relationships between two variables that exhibit a constant rate of change. In many real-world scenarios, the relationship between two quantities can be approximated as linear, making linear equations a valuable tool for analysis and prediction.
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Example 1: Distance and Time: The distance traveled by an object moving at a constant speed is linearly related to the time elapsed. If an object travels at a speed of 16 miles per hour, the equation relating distance (y) and time (x) can be expressed as y = 16x. This equation is similar in form to the equation we've been discussing, highlighting the applicability of linear equations in modeling motion.
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Example 2: Cost and Quantity: The total cost of purchasing a certain number of items at a fixed price per item is linearly related to the quantity purchased. If each item costs $16, the equation relating total cost (y) and quantity (x) can be expressed as y = 16x. This equation demonstrates the use of linear equations in economic modeling. — Cowboys Vs Rams: Detailed Stats And Historical Matchups
Solving Problems
Linear equations can be used to solve a variety of real-world problems. By setting up an equation that represents the problem's constraints and relationships, we can use algebraic techniques to find the solution.
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Example 3: Mixture Problems: Mixture problems involve combining two or more substances with different concentrations to obtain a mixture with a desired concentration. Linear equations can be used to determine the amounts of each substance needed to achieve the desired mixture.
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Example 4: Break-Even Analysis: In business, break-even analysis is used to determine the number of units that need to be sold to cover the fixed costs of production. Linear equations can be used to model the relationship between revenue, cost, and profit, allowing businesses to calculate the break-even point.
Applications in Different Fields
Linear equations find applications in a wide range of fields:
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Physics: Linear equations are used to describe motion, forces, and energy.
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Engineering: Linear equations are used in circuit analysis, structural analysis, and control systems.
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Economics: Linear equations are used in supply and demand analysis, cost-benefit analysis, and economic forecasting.
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Computer Science: Linear equations are used in computer graphics, data analysis, and machine learning.
Conclusion
The equation 5y + 16x = 4y + 10 serves as a gateway to understanding the fundamental principles of linear equations. Through simplification, isolating variables, and expressing 'y' in terms of 'x', we've unveiled the infinite solutions that lie along the line represented by this equation. Furthermore, we've explored the real-world applications of linear equations, highlighting their significance in modeling relationships and solving problems across diverse fields. By mastering the concepts and techniques presented in this article, readers can confidently tackle linear equations and apply them to a wide range of mathematical and practical challenges. — Magic Vs Bucks: Game Stats, Score, Top Performers
FAQ: Understanding Linear Equations
What is a linear equation?
A linear equation is a mathematical equation that represents a straight line when graphed on a coordinate plane. It involves variables raised to the power of 1 and can be written in various forms, such as slope-intercept form (y = mx + b) or standard form (Ax + By = C), where m, b, A, B, and C are constants. Linear equations are fundamental in algebra and have wide applications in various fields.
How do I solve a linear equation with two variables?
Solving a linear equation with two variables typically involves isolating one variable in terms of the other. This can be done by using algebraic manipulations such as combining like terms, adding or subtracting the same quantity from both sides, or multiplying or dividing both sides by the same non-zero quantity. Once one variable is isolated, you can substitute values for the other variable to find corresponding solutions.
Why does a linear equation like 5y + 16x = 4y + 10 have infinite solutions?
A linear equation with two variables has infinite solutions because for every value you choose for one variable (e.g., x), you can calculate a corresponding value for the other variable (e.g., y) that satisfies the equation. This creates a continuous set of solutions that can be represented as a straight line on a graph. Each point on the line represents a valid solution to the equation.
Can you provide a step-by-step example of solving 5y + 16x = 4y + 10?
Certainly! First, subtract 4y from both sides: 5y - 4y + 16x = 4y - 4y + 10, which simplifies to y + 16x = 10. Next, subtract 16x from both sides: y + 16x - 16x = 10 - 16x, resulting in y = 10 - 16x. This equation expresses y in terms of x, allowing you to find solutions by substituting different values for x.
In the equation y = 10 - 16x, what do the numbers 10 and -16 represent?
In the equation y = 10 - 16x, which is in slope-intercept form (y = mx + b), 10 represents the y-intercept (b), the point where the line crosses the y-axis. The number -16 represents the slope (m) of the line, indicating the rate of change of y with respect to x. In this case, for every 1 unit increase in x, y decreases by 16 units.
How are linear equations used in real-world applications?
Linear equations are used to model various real-world situations where there is a linear relationship between two variables. Examples include calculating the distance traveled at a constant speed over time, determining the total cost of purchasing a certain number of items at a fixed price, or modeling the relationship between supply and demand in economics. They are fundamental tools in science, engineering, economics, and more.
What is the significance of expressing 'y' in terms of 'x' in a linear equation?
Expressing 'y' in terms of 'x' (or vice versa) allows us to easily see the relationship between the two variables. It puts the equation in a form where we can readily determine how changes in 'x' affect 'y'. This is particularly useful for graphing the equation, finding solutions, and understanding the behavior of the linear relationship represented by the equation.
Where can I find more resources to learn about linear equations?
There are many resources available to learn more about linear equations. Khan Academy (https://www.khanacademy.org/) offers free video lessons and practice exercises on algebra topics, including linear equations. Textbooks on algebra and precalculus also provide comprehensive coverage of linear equations. Additionally, many websites and online forums dedicated to mathematics education can offer further assistance and explanations.
