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Chapter 7: Problem 76

Solve for the indicated variable. Solve for \(x: y=2 x+15 x\).

Short Answer

Expert verified
x = \frac{y}{17}

Step by step solution

01

Combine Like Terms

In the given equation \(y = 2x + 15x\), you can combine the like terms (the terms with \(x\)). Add the coefficients of \(x\) together: \(2x + 15x = 17x\). So the equation simplifies to \(y = 17x\).
02

Solve for x

Now that you have the equation \(y = 17x\), solve for \(x\) by dividing both sides by 17. This gives \(x = \frac{y}{17}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
In algebra, combining like terms is an essential skill. It allows us to simplify expressions by adding or subtracting terms that have the same variables raised to the same power. In the example equation, \( y = 2x + 15x \), the terms \( 2x \) and \( 15x \) are considered "like terms" because both involve the variable \( x \) raised to the first power. We can combine these terms by adding their coefficients, which are the numbers in front of the variables.
  • The coefficient of \( 2x \) is 2.
  • The coefficient of \( 15x \) is 15.
  • Adding them together gives 17, so \( 2x + 15x = 17x \).
So, when you encounter like terms, simply add or subtract their coefficients while keeping the variable part unchanged. This process results in a simplified expression like \( y = 17x \).
Simplifying Expressions
Simplifying an expression involves combining like terms to make it easier to understand and solve. Once we have identified and combined any like terms, the expression becomes simpler. In the given equation \( y = 17x \), after combining like terms, the expression is now in its simplest form as there are no further terms to combine.
  • Look for like terms that can be merged.
  • Use arithmetic operations to combine them.
  • The goal is to express the equation with as few terms as possible.
Simplifying expressions often makes the next steps, such as solving for a variable, more straightforward. This streamlined form, like \( y = 17x \), allows us to focus on isolating the desired variable.
Variable Isolation
Variable isolation is the process of "solving for a variable" to find its value in terms of the other numbers or variables given in the equation. After simplifying an expression, this process becomes easier. In our exercise, the equation \( y = 17x \) is ready for isolation. Here, we want to solve for \( x \), meaning we need \( x \) by itself on one side of the equation.
  • To isolate \( x \), perform inverse operations to cancel other coefficients or terms.
  • Since \( x \) is multiplied by 17, divide both sides of the equation by 17.
  • This results in \( x = \frac{y}{17} \).
The aim of variable isolation is to express one variable solely in terms of the other known values. This gives a clear formula, like \( x = \frac{y}{17} \), making it easy to calculate specific values when substituting \( y \).

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