Problem 62 - Elementary Algebra

Question

For Problems 55-70, solve each equation for the indicated variable. (Objective 4) $$ y=-7 x+10 \text { for } x $$

Step-by-Step Solution

Answer

\( x = -\frac{y - 10}{7} \)

Step 1: Identify the Equation

The equation provided is \( y = -7x + 10 \). Our goal is to solve this equation for the variable \( x \).

Step 2: Isolate the Term with x

Subtract 10 from both sides of the equation to isolate the terms involving \( x \) on one side:\[ y - 10 = -7x \]

Step 3: Solve for x

To solve for \( x \), divide both sides of the equation by -7 to isolate \( x \):\[ x = \frac{y - 10}{-7} \]

Step 4: Rearrange into a More Intuitive Form

Although \( x = \frac{y - 10}{-7} \) is correct, it may be clearer to express it as:\[ x = -\frac{y - 10}{7} \] or equivalently, \( x = -\frac{y}{7} + \frac{10}{7} \) to explicitly separate the terms.

Key Concepts

Variable Isolation

Variable isolation is crucial when solving equations. It means getting the variable you're solving for by itself on one side of the equation. This makes it possible to determine the value of that variable directly. For example, in the equation given, \( y = -7x + 10 \), we wanted to solve for \( x \).

To do this, you need to rearrange the equation so that \( x \) stands alone. This process often involves two important steps:

• Identify what needs to be moved from one side of the equation to the other.
• Use mathematical operations to make those movements possible.

In our exercise, we first subtracted 10 from both sides to shift all terms involving \( x \) to one side. This left us with \( y - 10 = -7x \). After this, we divided by -7 to get \( x \) by itself, giving us \( x = \frac{y - 10}{-7} \).

Doing these steps ensures that \( x \) is isolated, making the equation more manageable and easy to solve.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying an equation using different mathematical operations such as addition, subtraction, multiplication, and division. These operations help us to isolate the variable and, ultimately, solve the equation.

In the solved equation \( y = -7x + 10 \), we performed several algebraic manipulations:

• By subtracting 10 from both sides, we managed to keep the equation balanced. This is a basic principle in algebra: what you do to one side must be done to the other to maintain equality.
• Dividing both sides by -7 isolated \( x \), showing the impact of multiplication and division in reversing operations.

These manipulations may seem simple, but they are foundational. Being comfortable with such techniques is key to tackling more complex equations. Practicing these skills will improve your ability to handle various algebraic problems.

Linear Equations

Linear equations are equations of the first degree, which means they involve variables raised to the power of one. The standard form of a linear equation in two variables usually looks like \( Ax + By = C \).

Our original equation, \( y = -7x + 10 \), is considered linear. This is because it matches the slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Understanding this form helps to quickly identify the components of the equation.

The key characteristics of linear equations include:

• They graph as straight lines.
• Solutions are found using familiar operations like addition, subtraction, etc.
• They can be manipulated into different forms while still representing the same relationship.

Recognizing that our equation belongs to the linear family helps in predicting the outcome and understanding the relationships between the variables involved. Mastery of linear equations is essential in building a foundation for further algebraic learning.