Problem 34 - Elementary Algebra

Question

Solve. $$ -3 x+8-4 x+2=10 $$

Step-by-Step Solution

Answer

The solution is \(x = 0\).

Step 1: Combine Like Terms

The given equation is \(-3x + 8 - 4x + 2 = 10\).Combine the like terms, which are \(-3x\) and \(-4x\):\(-3x - 4x = -7x\).Thus, the equation becomes:\(-7x + 8 + 2 = 10\).

Step 2: Simplify the Equation

Now, simplify the constant terms on the left side of the equation:\(8 + 2 = 10\).So the equation simplifies to:\(-7x + 10 = 10\).

Step 3: Isolate the Variable Term

Subtract 10 from both sides of the equation to isolate the variable term:\(-7x + 10 - 10 = 10 - 10\).This results in:\(-7x = 0\).

Step 4: Solve for x

Divide both sides of the equation by \(-7\) to solve for \(x\):\(x = \frac{0}{-7}\).Since dividing zero by any non-zero number is zero, \(x = 0\).

Key Concepts

Like Terms in Linear Equations

When dealing with linear equations, it's important to identify and combine 'like terms.' But what exactly are like terms? They are terms in an equation that have the same variable raised to the same power. **For example:** - In the equation \( -3x + 8 - 4x + 2 \), the terms \( -3x \) and \( -4x \) are like terms because they both contain the variable \( x \) raised to the first power.By combining like terms, we simplify the equation, making it easier to solve. In our example:- \( -3x - 4x \) combines to \( -7x \).This process helps to reduce the number of terms in the equation, allowing us to proceed to the next steps with greater ease.

Isolating Variables in Equations

Isolating variables is a crucial step when solving equations. It means altering the equation so that the desired variable stands alone on one side. Let's see how this works in practice.Starting with our simplified equation:- \( -7x + 10 = 10 \).To isolate \( x \), we need to remove the constant term on the same side as \( x \). In our equation, this is 10:- **Subtract** 10 from both sides:\( -7x + 10 - 10 = 10 - 10 \).This gives us:- \( -7x = 0 \).Now, the variable term \( -7x \) is isolated, and you're one step closer to finding the value of \( x \). This process simplifies the equation, making the variable the focal point.

Solving for x in Linear Equations

Once the variable is isolated, the final step is solving for \( x \). This means finding the value of \( x \) that makes the equation true. Here's how it's done:From our equation:- \( -7x = 0 \).Our aim is to have \( x \) by itself, so we need to eliminate the coefficient of \( x \), which is \( -7 \). We do this by dividing both sides by \( -7 \):- \( x = \frac{0}{-7} \).Since zero divided by any non-zero number is zero, we find:- \( x = 0 \).This tells us that when \( x = 0 \), the original equation \(-3x + 8 - 4x + 2 = 10\) holds true. Solving for \( x \) in this manner is the heart of working with linear equations, as it reveals the specific value that satisfies the equation.