Problem 32 - Algebra for College Students

Question

Solve each equation for \(x\). \(x(a-b)=m(x-c)\)

Step-by-Step Solution

Answer

\(x = \frac{-mc}{a-b-m}\)

Step 1: Expand Both Sides

Start by expanding both sides of the equation. On the left-hand side, we have \(x(a-b)\). On the right-hand side, we have \(m(x-c)\). Expanding gives:\[xa - xb = mx - mc\]

Step 2: Isolate Terms with \(x\)

Next, we need to collect all terms containing \(x\) on one side of the equation. This requires moving terms as follows:\[xa - xb - mx = -mc\]

Step 3: Factor out \(x\)

We have terms with \(x\) on one side: \(xa - xb - mx\). Factor out \(x\) from these terms:\[x(a-b-m) = -mc\]

Step 4: Solve for \(x\)

To solve for \(x\), divide both sides of the equation by \(a-b-m\), provided \(a - b - m eq 0\):\[x = \frac{-mc}{a-b-m}\]

Key Concepts

Equation Expansion

Equation expansion is a crucial step in simplifying expressions and solving algebraic equations. It involves distributing or multiplying terms across brackets and opening them into simpler terms without parentheses. In our example, the original equation is given by:\[x(a-b) = m(x-c)\]To expand it, we apply the distributive property. This property says that for any numbers or expressions, \(a, b, c\), \(a(b+c) = ab + ac\). Applying it to both sides of our equation:

• Left side: \(x(a-b)\) expands to \(xa - xb\)
• Right side: \(m(x-c)\) expands to \(mx - mc\)

Thus, the expanded form of our equation is \(xa - xb = mx - mc\). This forms a simpler equation without parentheses, making further manipulation easier.

Factoring in Algebra

Factoring involves expressing an expression as a product of its factors. It's often used to simplify algebraic expressions or solve equations. After expanding the equation, we have:\[xa - xb - mx = -mc\]Here, we notice that the left-hand side has multiple terms that include the variable \(x\). Factoring in algebra allows us to simplify such expressions.When looking to factor, identify a common factor in each term. In this case, the common factor is \(x\), since it appears in all three terms:

• First term: \(xa = x \cdot a\)
• Second term: \(-xb = x \cdot (-b)\)
• Third term: \(-mx = x \cdot (-m)\)

By factoring \(x\) out of each term, the expression simplifies to:\[x(a-b-m)= -mc\]With \(x\) factored out, the equation becomes simpler and ready for the next step, isolating \(x\). Factoring is a powerful tool in algebra, especially in solving quadratic equations and polynomial functions.

Isolating Variables

Isolating variables is the process of rearranging an equation to get the unknown variable on one side. This technique helps in determining the variable's value directly. Let's isolate \(x\) from:\[x(a-b-m) = -mc\]To isolate \(x\), divide both sides of the equation by the coefficient \((a-b-m)\). The coefficient is the factor multiplied by \(x\):\[x = \frac{-mc}{a-b-m}\]This step is valid as long as \(a-b-m eq 0\). If the only terms on one side involve the variable you’re solving for, it simplifies computation. However, ensure the divisor is non-zero to avoid undefined operations. Isolating variables makes equations easier to interpret and solve, sidestepping complicated manipulation.