Problem 10 - Algebra 1: Concepts and Skills

Question

Decide whether the equation is true or false. Use the distributive property to explain your answer. $$ -2(4-3)=-8+6 $$

Step-by-Step Solution

Answer

The equation is true.

Step 1: Apply Distributive Property

Multiply -2 with each of the terms inside the bracket: \[-2(4-3) = -2*4 + (-2)*(-3) = -8 + 6 \]

Step 2: Perform Calculations on the Left Side of Equation

Calculate the result of -8 + 6 on the left side, which equals to -2.

Step 3: Perform Calculations on the Right Side of Equation

Calculate the result of -8 + 6 on the right side, which also equals to -2.

Step 4: Verify Equality

Both sides – left side (-2) and right side (-2) – are equal. Hence, the equation is true.

Key Concepts

Algebra

Algebra is a branch of mathematics that uses letters and symbols to represent numbers and quantities in equations and formulas. It allows us to perform calculations and solve problems even when we do not know the specific values of the variables involved. In simpler terms, algebra is like a puzzle where we use known quantities to find unknowns through logical operations.
For example, when you encounter an equation like \(-2(4-3) = -8 + 6\), algebra helps you decide whether these two expressions on either side of the equal sign are indeed equal.
Using algebraic techniques such as the distributive property, we can proceed to simplify and check both sides of the equation. This way, we can see whether the equality holds true or not. Algebra provides a systematic way to manipulate expressions and find solutions to mathematical problems.

Equations

Equations are mathematical statements that express the equality between two expressions. They are fundamental in algebra and are used to find unknown values by maintaining the balance between both sides of the equation. An equation often uses the equals sign \(=\) to show this balance.
Here is what an equation looks like: when you have something like \(-2(4-3) = -8 + 6\), the goal is to find out if both sides of the equation are equivalent. Breaking it down, the left-hand side can be simplified using arithmetic operations, while the right-hand side provides a separate calculation to compare.
The key process in working with equations involves:

• Identifying terms on both sides
• Simplifying expressions
• Ensuring both sides balance to establish if the equation is true or false

This systematic procedure is vital in confirming equations' validity, using properties like distribution effectively.

Mathematical Proof

Mathematical proof is a logical argument that demonstrates the truth of a given proposition. This involves using known facts, definitions, and previously established results to show conclusively that a statement is correct. It plays a crucial role in mathematics by ensuring that conclusions are founded on solid reasoning.
In the context of the equation \(-2(4-3) = -8 + 6\), mathematical proof involves using the distributive property to reach a logical conclusion from both sides of the equation. By distributing \(-2\) through the terms in the parentheses, you simplify the expression precisely:

• On simplifying the left side: \(-2 * 4 + (-2) * (-3) = -8 + 6\)
• Calculate: \(-8 + 6 = -2\)

The steps create a logical flow to show equality, thus proving that the equation holds true. The use of mathematical proof ensures that each step is clear and justified, verifying the truth of equations through demonstrated reasoning.