Problem 61 - Elementary Algebra

Question

For the following problems, solve the literal equations for the indicated variable. When directed, find the value of that variable for the given values of the other variables. Solve \(-9 x+3 y+15=\) 0for \(y\).

Step-by-Step Solution

Answer

Question: Solve the linear equation -9x + 3y + 15 = 0 for y in terms of x. Answer: y = 3x - 5

Step 1: Rewrite the given equation

We will first rewrite the given linear equation, \(-9x + 3y + 15 = 0\) : $$-9x + 3y + 15 = 0$$

Step 2: Isolate y

To isolate \(y\) on one side of the equation, add \(9x\) and subtract \(15\) from both sides of the equation: $$3y = 9x - 15$$

Step 3: Solve for y

Now, we need to solve for \(y\). To do this, divide both sides of the equation by 3: $$y = \frac{9x - 15}{3}$$

Step 4: Simplify the expression

We can further simplify this expression by dividing the numerator by 3: $$y = 3x - 5$$ Now we have the value of \(y\) in terms of \(x\). The final expression for \(y\) is: $$y = 3x - 5$$

Key Concepts

Linear Equations

Linear equations are mathematical expressions that represent a straight line when graphed. These equations involve variables that have a degree of one, meaning they are not raised to any power other than one. For example, in the equation \(-9x + 3y + 15 = 0\), both \(x\) and \(y\) are raised to the first power, making it a linear equation.

The general form of a linear equation in two variables is \(Ax + By + C = 0\) where \(A\), \(B\), and \(C\) are constants. This particular form allows one to easily recognize an equation as linear. Linear equations can have one solution, no solution, or infinitely many solutions depending on how they intersect when graphed. Understanding linear equations is crucial as they are foundational in understanding more complex algebraic concepts. They are often the first types of equations students encounter in algebra, and mastering them is essential for solving real-world problems.

Isolation of Variables

Isolation of variables is a key technique used in algebra to solve equations. It involves rearranging the equation so that one variable is alone on one side. By isolating a variable, we are explicitly finding its value in terms of other variables or constants. For instance, in our equation \(-9x + 3y + 15 = 0\), we isolated \(y\) by adding \(9x\) and subtracting \(15\):

\(3y = 9x - 15\).

This step is crucial because it converts a complex equation into something manageable. The main goal is to have the variable of interest by itself, which at times necessitates performing inverse operations. These operations usually involve addition, subtraction, multiplication, or division. The order of these operations can be significant, and understanding how to reverse operations helps students break down complex equations into simpler parts.

Algebraic Manipulation

Algebraic manipulation refers to the process of altering an expression or equation using various algebraic rules and techniques. This often involves simplifying expressions, combining like terms, and performing arithmetic operations such as addition, subtraction, multiplication, and division. In the equation \(3y = 9x - 15\), we perform algebraic manipulation by dividing each term by 3:

\(y = \frac{9x - 15}{3}\).

Algebraic manipulation is integral to solving for variables and simplifying equations, making them easier to interpret and solve. Deftly manipulating algebraic expressions allows you to transform complicated problems into straightforward solutions. This is particularly helpful not only in academic settings but also in solving practical problems related to finance, science, engineering, and everyday life situations where relationships between variables need to be established and interpreted clearly.