Problem 35 - Elementary Algebra

Question

Solve the equation \(2 y=5(3 x+7)\) if \(x=-1\).

Step-by-Step Solution

Answer

Answer: When x = -1, the value of y is 10.

Step 1: Substitute the given value of x into the equation

Replace x with -1 in the equation \(2 y=5(3 x+7)\): \(2y = 5(3(-1) + 7)\)

Step 2: Simplify the expression inside the parentheses

Perform the operation inside the parentheses: \(2y = 5(-3 + 7)\)

Step 3: Continue simplifying the expression

Simplify the expression inside the parentheses and multiply by the constant 5: \(2y = 5(4)\)

Step 4: Multiply the constants

Perform the multiplication: \(2y = 20\)

Step 5: Solve for y

Divide both sides of the equation by 2 to isolate y: \(y = \frac{20}{2}\)

Step 6: Calculate the value of y

Perform the division to find the value of y: \(y = 10\) The solution of the equation \(2 y=5(3 x+7)\) when \(x=-1\) is \(y=10\).

Key Concepts

Algebraic Substitution

Algebraic substitution is a method used in solving equations where you replace a variable with a given numerical value or another expression. In our example, we are given the equation \(2y = 5(3x + 7)\) and a specific value for \(x\) which is \(-1\). To proceed with algebraic substitution, we replace every occurrence of \(x\) in the equation with \(-1\).
This results in the expression \(2y = 5(3(-1) + 7)\). Substitution simplifies the problem because it transforms a general equation into one that is specific and easier to solve by using known quantities.

• Identify the given value(s).
• Replace the variable with the given value(s) in the equation.

By following these simple steps, you can tackle equations with more confidence and ease.

Linear Equations

Linear equations are fundamental in algebra, and they form the basis for more complex topics in mathematics. A linear equation can be recognized by its structure which features variables raised to the power of one (i.e., no squares, cubes, etc.) spread over a straight line. In our equation \(2y = 5(3x + 7)\), the key linear component is straightforward, represented by the first power of \(x\) and \(y\). Linear equations typically show relationships where the solution set forms a line when graphed on a coordinate plane.
The goal is to isolate the variable in question, in this case \(y\), through operations such as addition, subtraction, multiplication, and division. Here, the equation was manipulated to first substitute \(x\) and then solve for \(y\).
Knowing how to handle linear equations is crucial for solving numerous real-world problems, such as calculating distance, cost, or any situation involving a constant rate of change.

Simplification Process

The simplification process involves reducing an equation or expression to its simplest form while maintaining its equality or original meaning. This is achieved through basic arithmetic operations and often makes the solving process more approachable.
For the equation \(2y = 5(3(-1) + 7)\), the work begins by simplifying inside the parentheses, performing the operations in a step-by-step manner:

• Calculate inside the parentheses: \(-3 + 7 = 4\).
• Multiply by 5: \(5 \times 4 = 20\).
• Divide by 2 to solve for \(y\): \(y = \frac{20}{2} = 10\).

Simplification makes the equation manageable by reducing steps and preventing errors. It is a critical process that aids in deriving the solution effectively and efficiently. Learning to simplify expressions is a vital skill that extends beyond mathematics into everyday problem-solving and decision-making.