Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 2(x-4)+3(x+5)=2x-2

Verified step by step guidance

1. Start by applying the distributive property to both sides of the equation: expand \$2(x-4)\( and \)3(x+5)\( to get \)2 \cdot x - 2 \cdot 4 + 3 \cdot x + 3 \cdot 5$.
2. Simplify the expressions on the left side by performing the multiplications: this results in \$2x - 8 + 3x + 15$.
3. Combine like terms on the left side: add \$2x\( and \)3x\( to get \)5x\(, and combine \)-8\( and \)15\( to get \)7\(, so the left side becomes \)5x + 7$.
4. Rewrite the equation with the simplified left side and the right side as is: \$5x + 7 = 2x - 2$.
5. To isolate \(x\), subtract \$2x\( from both sides and subtract \)7\( from both sides, resulting in \)5x - 2x = -2 - 7\(, which simplifies to \)3x = -9\(. From here, you can solve for \)x$ by dividing both sides by 3.

Key Concepts

Solving Linear Equations

Solving linear equations involves isolating the variable on one side of the equation using inverse operations such as addition, subtraction, multiplication, and division. The goal is to find the value of the variable that makes the equation true. For example, simplifying and combining like terms helps to reduce the equation to a simpler form.

Types of Equations: Identity, Conditional, and Inconsistent

An identity is true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solution. After solving, if the equation simplifies to a true statement like 0=0, it is an identity; if it results in a false statement like 0=5, it is inconsistent.

Distributive Property

The distributive property allows you to multiply a single term by each term inside parentheses, expressed as a(b + c) = ab + ac. This property is essential for expanding expressions before combining like terms and simplifying equations, as seen in 2(x-4) + 3(x+5).