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Chapter 13: Problem 45

Solve each equation. Check your solution. $$30=3+y$$

Short Answer

Expert verified
The solution is \(y = 27\).

Step by step solution

01

Isolate the Variable

To solve the equation \(30 = 3 + y\), we need to isolate the variable \(y\). Start by subtracting \(3\) from both sides of the equation:\[30 - 3 = 3 + y - 3\]This simplifies to:\[27 = y\]
02

Solution Verification

To verify that \(y = 27\) is correct, substitute \(27\) back into the original equation:\[30 = 3 + y\]\[30 = 3 + 27\]\[30 = 30\]Since both sides of the equation are equal, \(y = 27\) is indeed the correct solution.

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Key Concepts

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

Algebraic Expressions
Algebraic expressions can sometimes seem intimidating, but they are a crucial part of solving equations. An algebraic expression is a mathematical phrase that can include numbers, operations, and variables. In our problem, the expression is represented by \(3 + y\), where 3 is a constant and \(y\) is the variable that can take on different values. Understanding algebraic expressions is key to solving equations for several reasons:
  • They provide a way to express relationships between different quantities.
  • By manipulating algebraic expressions, we can solve for unknown variables.
  • They allow us to apply mathematical operations to find solutions to problems.
Breaking down a problem into its algebraic components and simplifying each part step by step can turn a complex task into a more manageable one. This is what makes algebra a powerful tool in mathematics.
Isolate the Variable
One of the core steps in solving equations is to isolate the variable. This means rearranging the equation so that the variable stands alone on one side of the equation, usually the left. In our example, the original equation is \(30 = 3 + y\). To isolate \(y\), we need to undo the operations that are combined with \(y\).Here's how you can do it:
  • Identify any constants added or subtracted from the variable \(y\). In our equation, \(3\) is added to \(y\).
  • Perform the inverse operation to both sides of the equation to remove this constant. Since \(3\) is added to \(y\), we subtract \(3\) from both sides, leading to \(30 - 3 = 3 + y - 3\).
  • Simplify both sides to see that \(27 = y\), thus isolating \(y\).
Isolating the variable lets you easily see the value that solves the equation. It transforms a complex equation into a straightforward answer, making this a vital skill in algebra.
Solution Verification
Once you've isolated the variable and found a solution, it's important to verify that the solution is correct. This can be done by plugging the solution back into the original equation and checking whether both sides of the equation are equal. In our example, the solution was \(y = 27\).To verify:
  • Substitute \(27\) back into the original equation, replacing \(y\).
  • This gives us: \(30 = 3 + 27\).
  • Perform the addition on the right side: \(3 + 27 = 30\).
Since both sides of the equation equal 30, our solution \(y = 27\) is verified as correct.Verifying your solution is critical as it confirms that your initial equation manipulation was accurate. It boosts confidence and ensures error-free math practice.

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Most popular questions from this chapter

The graph of quadratic functions may have one maximum or one minimum point. The maximum point of a graph is the point with the greatest \(y\) -value coordinate. The minimum point is the point with the least \(y\) -value coordinate. Graph each equation. Find the coordinates of each point. the maximum point of the graph of \(y=-x^{2}+7\)

Identify the equation that doesn't belong with the other three. A. \(y=\frac{1}{3} x-x\) B. \(y^{2}=3 x\) C. \(5 x+y=6\) D. \(x=5 y\)

Find each sum. $$\left(5 x^{2}-7 x+9\right)+\left(3 x^{2}+4 x-6\right)$$

Find each product. $$3 y(8-7 y)$$

Use newspapers, magazines, or the Internet to find real world examples of nonlinear situations.
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