Problem 77 - Algebra for College Students

Question

Solve each equation, and check the solution. \(0.05 x+0.10(200-x)=0.45 x\)

Step-by-Step Solution

Answer

x = 40

Step 1 - Expand the Equation

Start by expanding the equation \[0.05 x + 0.10(200 - x) = 0.45 x\].Distribute the 0.10 through the parentheses:\[0.05 x + 20 - 0.10 x = 0.45 x\].

Step 2 - Simplify Like Terms

Combine the like terms on the left side of the equation:\[0.05 x - 0.10 x + 20 = 0.45 x\].This simplifies to:\[-0.05 x + 20 = 0.45 x\].

Step 3 - Isolate the Variable x

Move all terms containing x to one side of the equation and the constant terms to the other side. Add 0.05x to both sides to isolate x:\[20 = 0.45 x + 0.05 x\].Combine the x terms:\[20 = 0.50 x\].

Step 4 - Solve for x

Divide both sides by 0.50 to solve for x:\[x = \frac{20}{0.50} = 40\].

Step 5 - Check the Solution

Substitute the solution back into the original equation to verify it:\[0.05(40) + 0.10(200 - 40) = 0.45(40)\].Calculate each term:\[2 + 0.10(160) = 18\].This simplifies to:\[2 + 16 = 18\], which confirms the solution is correct.Therefore, the solution is \(x = 40\).

Key Concepts

linear equations

Linear equations are equations that form a straight line when graphed. They are typically in the form: \(ax + by = c\). Each term can include a constant or a product of a constant and a variable. \ Linear equations have only one or two variables, with no exponents or powers. \ For example, in our exercise, we have \( 0.05x + 0.10(200 - x) = 0.45x \).

isolate variable

Isolating the variable in an equation means getting the variable alone on one side of the equation. This helps us to find the value of the variable. \ To isolate the variable, follow these steps: \

\
• Move terms containing the variable to one side. \
• Move constant terms to the other side. \
• Combine like terms if possible. \
• Finally, divide or multiply to solve for the variable. \

\ In our problem, we isolated \( x \) by first moving the \( -0.05x \) term and then combining like terms to get \( x = 40 \).

combining like terms

Combining like terms helps to simplify an equation. Like terms have the same variable raised to the same power. \ For instance, \( 5x \) and \( 2x \) are like terms, but \( 5x \) and \( 2y \) are not. \ In our example, we combined \( 0.05x \) and \( -0.10x \) to get \( -0.05x. \) This simplification step is crucial as it makes the equation easier to solve.

distribution in algebra

Distribution is an algebraic technique used to multiply a single term by each term within parentheses. This is known as the distributive property: \( a(b + c) = ab + ac \). \ In our problem, we used distribution to expand \( 0.10(200 - x) \) into \( 0.10(200) - 0.10(x) \). \ After distribution, the equation was easier to handle, leading us to the solution more efficiently.