Solving For Y A Step-by-Step Guide To Rearranging -5x - 4y = -8
In mathematics, we often encounter equations involving multiple variables. Sometimes, we need to rearrange these equations to express one variable in terms of the others. This process is particularly useful when we want to analyze the relationship between variables, create graphs, or solve systems of equations. In this article, we will focus on rearranging a linear equation to make x the independent variable and express y as a function of x. We'll break down the steps involved and provide a clear, comprehensive explanation to guide you through the process. Understanding how to isolate variables is a fundamental skill in algebra and calculus, and it lays the groundwork for more advanced mathematical concepts. This guide will not only help you solve the given equation but also equip you with the tools to tackle similar problems in the future. Whether you're a student learning algebra or simply looking to refresh your math skills, this article will provide a valuable resource.
The Given Equation
Let's start with the equation we want to rearrange:
-5x - 4y = -8
Our goal is to isolate y on one side of the equation, expressing it in terms of x. This means we want to rewrite the equation in the form y = f(x), where f(x) is an expression involving x. Rearranging equations is a crucial skill in algebra, as it allows us to understand the relationship between variables and to solve for unknowns. The given equation, -5x - 4y = -8, is a linear equation in two variables, x and y. Linear equations are fundamental in mathematics and have numerous applications in various fields, including physics, economics, and computer science. Understanding how to manipulate these equations is essential for solving real-world problems. The process of isolating y involves several algebraic steps, each of which must be performed carefully to maintain the equality of the equation. By following these steps, we can transform the original equation into an equivalent form that explicitly expresses y as a function of x. This form is particularly useful for graphing the equation, as it allows us to easily determine the y-value for any given x-value. Furthermore, rearranging equations is a key technique in solving systems of equations, where we need to find the values of variables that satisfy multiple equations simultaneously. This skill is not only important for academic success but also for practical applications in various professional fields.
Step 1: Isolate the Term with y
The first step is to isolate the term containing y on one side of the equation. In this case, the term is -4y. To do this, we need to eliminate the term -5x from the left side. We can achieve this by adding 5x to both sides of the equation. Remember, adding the same value to both sides of an equation maintains the equality.
-5x - 4y + 5x = -8 + 5x
This simplifies to:
-4y = 5x - 8
Isolating the term with y is a critical step in solving for y. It involves using the addition property of equality, which states that adding the same quantity to both sides of an equation does not change the equality. This property is a fundamental principle in algebra and is used extensively in solving equations. By adding 5x to both sides of the original equation, we effectively cancel out the -5x term on the left side, leaving the -4y term isolated. This step prepares the equation for the next operation, which involves dividing both sides by the coefficient of y. It's important to perform this step accurately, as any error here will propagate through the rest of the solution. The goal is to manipulate the equation in a way that y is the only term on one side, allowing us to express y in terms of x. This process is not only useful for solving equations but also for understanding the relationship between variables. By isolating y, we can see how changes in x affect the value of y. This understanding is crucial in various applications, such as graphing linear equations and modeling real-world phenomena.
Step 2: Solve for y
Now that we have -4y = 5x - 8, we need to solve for y. To do this, we divide both sides of the equation by the coefficient of y, which is -4.
(-4y) / -4 = (5x - 8) / -4
This gives us:
y = (5x - 8) / -4
Solving for y involves dividing both sides of the equation by the coefficient of y. This step utilizes the division property of equality, which states that dividing both sides of an equation by the same non-zero quantity does not change the equality. In this case, we divide by -4, which is the coefficient of y. It's crucial to divide every term on the right side of the equation by -4 to maintain the equality. This step effectively isolates y on the left side, expressing it in terms of x. The resulting equation, y = (5x - 8) / -4, shows how y varies with x. This form is particularly useful for graphing the equation and for making predictions about the relationship between x and y. For example, we can easily substitute different values of x into the equation to find the corresponding values of y. This process is fundamental in many areas of mathematics and science, including calculus, physics, and engineering. Furthermore, understanding how to solve for a specific variable is essential for solving systems of equations, where we need to find the values of variables that satisfy multiple equations simultaneously. The ability to manipulate equations and isolate variables is a core skill in mathematics and is essential for success in higher-level courses.
Step 3: Simplify the Equation (Optional)
The equation y = (5x - 8) / -4 can be simplified further. We can divide each term in the numerator by -4:
y = (5x / -4) - (8 / -4)
This simplifies to:
y = -5/4x + 2
Simplifying the equation is an optional but highly recommended step. It makes the equation easier to understand and work with. In this case, we simplify y = (5x - 8) / -4 by dividing each term in the numerator by -4. This process involves the distributive property of division over addition and subtraction. We divide 5x by -4 to get -5/4x, and we divide -8 by -4 to get +2. The resulting equation, y = -5/4x + 2, is in slope-intercept form, which is a standard form for linear equations. This form makes it easy to identify the slope and y-intercept of the line. The slope is the coefficient of x, which is -5/4, and the y-intercept is the constant term, which is 2. The slope-intercept form is particularly useful for graphing linear equations, as the slope and y-intercept provide key information about the line's position and orientation in the coordinate plane. Furthermore, this form makes it easier to compare different linear equations and to analyze their properties. Simplifying equations is a fundamental skill in algebra and is essential for solving more complex problems. It not only makes the equations easier to work with but also provides insights into their underlying structure and relationships. In many cases, a simplified equation can reveal patterns and properties that are not immediately apparent in the original form.
Final Answer
Therefore, the equation -5x - 4y = -8 rearranged so that x is the independent variable is:
y = -5/4x + 2
The final answer, y = -5/4x + 2, represents the equation rearranged such that y is expressed in terms of x. This form is known as the slope-intercept form of a linear equation, where the coefficient of x (-5/4) represents the slope of the line, and the constant term (2) represents the y-intercept. The slope-intercept form is particularly useful for graphing the equation, as it provides direct information about the line's steepness and where it crosses the y-axis. Furthermore, this form makes it easy to compare different linear equations and to analyze their properties. Understanding how to rearrange equations into slope-intercept form is a fundamental skill in algebra and is essential for solving various problems involving linear relationships. For example, we can use this form to determine the equation of a line given its slope and y-intercept, or to find the point where two lines intersect. The process of rearranging equations is a key technique in mathematics and is used extensively in higher-level courses such as calculus and differential equations. It allows us to manipulate equations to gain insights into the relationships between variables and to solve for unknowns. In this case, by rearranging the equation -5x - 4y = -8, we have successfully expressed y as a function of x, which is a crucial step in understanding the behavior of the linear relationship represented by the equation.
In this article, we have demonstrated how to rearrange the equation -5x - 4y = -8 to express y in terms of x. This process involves isolating the y term and then solving for y. The final equation, y = -5/4x + 2, represents the same relationship as the original equation but with y as the dependent variable and x as the independent variable. This skill is crucial in algebra and beyond, as it allows us to analyze and manipulate equations to solve for unknowns and understand the relationships between variables. Mastering this technique will be invaluable as you progress in your mathematical studies.
