Problem 39 - Algebra 1

Question

Use a graphing calculator to find the solution of the equation. Check your solution algebraically. $$-1.6(1.5 x+7.5)=0.6(6 x+30)$$

Step-by-Step Solution

Answer

The solution to the equation \( -1.6(1.5 x+7.5)=0.6(6 x+30) \) is \( x = -5 \), as checked algebraically, the left side equals to the right side, verifying the solution as correct.

Step 1: Simplify both sides of the equation

Let's first distribute the factors \( -1.6 \) and \( 0.6 \) in their respective parentheses. For the left-hand side: \( -1.6(1.5x+7.5) = -2.4x - 12 \). For the right-hand side: \( 0.6(6x+30) = 3.6x + 18 \). The equation thus becomes: \(-2.4x - 12 = 3.6x + 18 \)

Step 2: Collect like terms

Firstly, bring together all the \(x\) terms on one side and keep constants on the other. In order to do that, add \( 2.4x \) to both sides, and subtract \( 18 \) on both sides. This will give you: \(6x = -30\)

Step 3: Solve for the unknown variable \(x\)

To isolate \(x\) and find its value, divide both sides by the coefficient of \(x\), which is \(6\) in this case. This gives: \( x = -30 / 6 \), thus \( x = -5 \) . You have now found the value of x using algebra.

Step 4: Check the Solution

Checking the solution involves substituting the derived value of \(x\) in the original equation and see if left side equals to the right side. Substituting \( x = -5 \) in the original equation gives: \( -1.6(1.5(-5)+7.5) =?= 0.6(6(-5)+30) \). Simplifying both sides gives: \( -12 =?= -12 \), which satisfies the equation.

Key Concepts

Algebraic Solutions

When solving algebraic equations, the goal is to find the value of the variable that makes the equation true. In the exercise given, we start by simplifying both sides of the equation. Distributing the coefficients (\(-1.6\) and \(0.6\)) within each set of parentheses transforms the equation into a simpler form. Once we accomplish this, we collect like terms. This involves moving all terms involving the variable to one side of the equation and isolating the constant terms on the other. Breaking down each step helps us understand how simplification works and prepares us for solving any other algebraic equation.
The final step is solving for the unknown variable, in this case, \(x\). We do this by dividing both sides of the equation by the coefficient of \(x\), giving us the solution \(x = -5\). Once the algebraic solution is found, it needs to be checked to ensure it's correct. We substitute the value of \(x\) back into the original equation to verify that both sides are equal. This not only helps confirm the calculation is correct but also enhances confidence in our algebraic manipulation skills.

Equation Solving

Equation solving is a fundamental aspect of algebra where the objective is to find the value of an unknown variable that satisfies the equation. This problem-solving process usually involves a series of sequential steps such as simplification, isolating variables, and performing operations to both sides of the equation, maintaining a balance.

• Simplification: Start by distributing any factors and combining like terms. This reduces complexity and aids in easier manipulation.
• Isolation: Move terms from one side to the other to gather all variable terms on one side and constants on the other. This often involves adding or subtracting terms from both sides.
• Solving: Solve for the variable by performing necessary arithmetic operations. These might be division, multiplication or sometimes additional simplifications.

In doing so, you turn complex expressions into solvable linear equations, as seen in the exercise. Equation solving can be practiced with numerous linear or non-linear problems, increasing overall mathematical aptitude and analytical skills.

Using Technology in Math Education

Modern technology, especially graphing calculators, plays a crucial role in math education by offering powerful means for solving complex equations quickly and accurately. In the context of the provided exercise, using a graphing calculator allows visual verification of the solution. These devices not only compute, but also graph functions, providing a comprehensive understanding of equations' behaviors.

• Efficiency: Graphing calculators speed up computations and provide immediate results, encouraging students to tackle more challenging problems.
• Visualization: They offer a visual representation of equations, helping students understand the nature of solutions through graphs, intersections, and slopes.
• Validation: By cross-checking algebraic solutions graphically, students can confirm correctness and understand the method's soundness.

Integration of technology in learning fosters a deeper engagement and interaction with mathematical concepts, aligning traditional methods with innovative tools. This makes learning math a more dynamic and insightful experience.