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Solving by Distributing Decimals and Collecting Terms
To solve , both sides contain a decimal factor multiplying a binomial, so the first step is to distribute on each side before collecting variable and constant terms.
Step 1 — Distribute on both sides: Multiply by each term inside the left parentheses and by each term inside the right parentheses. On the left: and . On the right: and . The equation becomes:
Step 2 — Collect variable terms on one side: Since , designate the left as the variable side. Subtract from both sides using the Subtraction Property of Equality:
Step 3 — Collect constant terms on the other side: Subtract from both sides:
Step 4 — Divide both sides by : Apply the Division Property of Equality to isolate :
Step 5 — Check by substitution: Replace with in the original equation:
Because both sides are equal, is confirmed as the correct solution. This example illustrates that when both sides of an equation contain a decimal factor multiplying a parenthesized expression, the distributive property must be applied to each side independently before the standard collecting technique can begin. Multiplying a decimal by a whole number — such as — may produce a whole-number coefficient, which simplifies the subsequent algebra.
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Ch.2 Solving Linear Equations and Inequalities - Elementary Algebra @ OpenStax
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Solving by Distributing Decimals and Collecting Terms
In a professional setting, such as a logistics manager calculating the number of shipping containers needed to meet a specific budget, you must solve linear equations systematically. Arrange the following steps of the standard four-step strategy for solving a linear equation in the correct order from start to finish.
A payroll specialist is using a standard four-step strategy to solve a linear equation to determine a staff member's total compensation. Match each step of the strategy with the correct action required.
A small business owner is solving a linear equation to determine the number of units they must sell to reach a break-even point. According to the standard four-step strategy for solving linear equations, what is the fourth and final step the owner should take?
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A logistics coordinator is solving a linear equation to determine a warehouse's reorder point for safety stock. According to the systematic four-step strategy for solving linear equations, the step that involves using the Addition or Subtraction Properties of Equality to move terms so that the variable remains alone on one side is called __________ the variable.
A facilities manager is using the systematic four-step strategy to solve a linear equation representing a warehouse's monthly energy consumption. True or False: According to this strategy, the final step is to simplify the final expressions through arithmetic operations to find the precise numerical value of the variable.
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A production supervisor is using the systematic four-step strategy to solve a linear equation that models the daily output of a manufacturing line. According to this strategy, which of the following best describes the actions the supervisor should take during the first step, 'Simplify both sides'?
A budget analyst is using the systematic four-step strategy to solve a linear equation representing a department's quarterly spending. After the analyst has successfully isolated the variable, what is the primary objective of the next step, 'Simplify the final expressions'?
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Learn After
A logistics manager is using the equation 0.36(100n + 5) = 0.6(30n + 15) to calculate the distribution of shipping costs between two regional hubs. To begin solving for the variable n, which mathematical property must be applied first to both sides of the equation to remove the parentheses?
A project manager is solving the equation 0.36(100n + 5) = 0.6(30n + 15) to determine the project duration 'n'. Arrange the steps below in the correct order to solve for n.
A logistics manager is comparing two different fuel efficiency models using the equation , where is the number of delivery trucks in the fleet. To begin solving the equation, the manager must distribute the decimals. Match each part of the equation with its correct value after this first step of distribution is completed.
A financial analyst is verifying the solution n = 0.4 for the budget balancing equation 0.36(100n + 5) = 0.6(30n + 15). True or False: If the solution is correct, substituting 0.4 back into the original equation will result in both sides simplifying to a value of 16.2.
A technician is using the equation 0.36(100n + 5) = 0.6(30n + 15) to determine the number of hours (n) required for an assembly task. After applying the distributive property to the right side of the equation, the resulting constant term (the product of 0.6 and 15) is ____.
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An operations analyst is solving the equation 0.36(100n + 5) = 0.6(30n + 15) to optimize a supply chain route. After simplifying the equation to the step 18n = 7.2, which mathematical property must be recalled and applied to both sides to isolate the variable n?
A workforce analyst is using the equation to balance staffing levels () between two departments. After distributing the decimals and collecting the variable terms, the analyst simplifies the equation to:
According to the standard algebraic procedure, which of the following is the next step to collect the constant terms on one side of the equation?