Question

Solve: \(4 x-5(2 x-1)=4-7(x+1)\)

Short answer

The solution to the equation is \(x = -8\).

Step-by-step solution

Distribute the multiplication

Distribute the multiplication through the parentheses on both sides of the equation. This means applying the multiplication to each term inside the parentheses:For the left-hand side: \(4x - 5(2x - 1) = 4x - 10x + 5\)For the right-hand side: \[4 - 7(x + 1) = 4 - 7x - 7\]

Simplify the equation

Combine like terms on both sides of the equation:\(4x - 10x + 5 = 4 - 7x - 7\)Which simplifies to:\(-6x + 5 = -7x - 3\)

Move all x terms to one side

Add \(7x\) to both sides to get all the x terms on one side of the equation:\(-6x + 7x + 5 = -7x + 7x - 3\)Simplifies to:\(x + 5 = -3\)

Solve for x

Subtract 5 from both sides to isolate x:\(x + 5 - 5 = -3 - 5\)This results in:\(x = -8\)

Key concepts

Distributive Property

The distributive property is a key principle in algebra that allows you to simplify equations by distributing multiplication over addition or subtraction within parentheses. It's expressed as: \[a(b + c) = ab + ac\]In our exercise, we apply the distributive property to expand both sides of the equation. For the left-hand side: \[4x - 5(2x - 1)\]Distributing \-5 over \(2x - 1\), we get:\[4x - 10x + 5\]Then, for the right-hand side, we have:\[4 - 7(x + 1)\]Distributing \-7 over \(x + 1\), we obtain:\[4 - 7x - 7\]Using the distributive property simplifies our equation, making the next steps easier.

Combining Like Terms

Combining like terms is essential when simplifying an equation. Like terms have the same variable raised to the same power. For example, \(4x\) and \(-10x\) are like terms because they both contain the variable \(x\).In our equation, after applying the distributive property, we had:\[4x - 10x + 5 = 4 - 7x - 7\]Next, we combine the like terms on both sides to streamline the equation. On the left-hand side, we combine \(4x\) and \(-10x\):\[4x - 10x = -6x\]Adding the constant \(5\) gives us:\[-6x + 5\]For the right-hand side, we combine the constants \(4\) and \(-7\):\[4 - 7 = -3\]Including \(-7x\), we get:\[-7x - 3\]This simplifies the equation to a more manageable form:\[-6x + 5 = -7x - 3\]

Isolating Variables

Isolating the variable is a crucial step to solving equations. The goal is to get the variable on one side of the equation and the constants on the other. After combining like terms, our equation is:\[-6x + 5 = -7x - 3\]To isolate \(x\), we should move all \(x\)-terms to one side. By adding \(7x\) to both sides, we neutralize the \(-7x\) on the right:\[-6x + 7x + 5 = -7x + 7x - 3\]Which simplifies to:\[x + 5 = -3\]Now, to fully isolate \(x\), we subtract \(5\) from both sides:\[x + 5 - 5 = -3 - 5\]This results in:\[x = -8\]The variable \(x\) is now isolated, giving us the solution.

Linear Equations

Linear equations are equations of the first degree, meaning they involve only linear terms (terms where the variable is raised to the power of one). They take the general form of:\[ax + b = c\]Where \(a\), \(b\), and \(c\) are constants. Solving a linear equation involves simplifying and isolating the variable. Our original problem:\[4 x - 5(2x - 1) = 4 - 7(x + 1)\]By distributing, combining like terms, and isolating the variable, we find:\[x = -8\]Linear equations are fundamental in algebra and understanding how to solve them is critical for more advanced math concepts. Focus on mastering the distributive property, combining like terms, and isolating variables, as they are the stepping stones to solving linear equations efficiently.