What are the possible steps involved in solving the equation shown? Select three options.
[tex]$3.5+1.2(6.3-7 x)=9.38$[/tex]
A. Add 3.5 and 1.2.
B. Distribute 1.2 to 6.3 and [tex]$-7 x$[/tex].
C. Combine 6.3 and [tex]$-7 x$[/tex].
D. Combine 3.5 and 7.56.
E. Subtract 11.06 from both sides.
Answer (1)
Distribute 1.2 to both terms inside the parenthesis: 3.5 + 1.2 ( 6.3 ) + 1.2 ( − 7 x ) = 3.5 + 7.56 − 8.4 x .
Combine the constant terms: 3.5 + 7.56 = 11.06 , resulting in 11.06 − 8.4 x = 9.38 .
Subtract 11.06 from both sides: − 8.4 x = 9.38 − 11.06 = − 1.68 .
The possible steps involved in solving the equation are: Distribute 1.2 to 6.3 and − 7 x , Combine 3.5 and 7.56, and Subtract 11.06 from both sides.
Explanation
Analyze the equation We are given the equation 3.5 + 1.2 ( 6.3 − 7 x ) = 9.38 and asked to identify the correct steps to solve for x . Let's analyze the equation and the possible steps.
Distribute First, we need to distribute the 1.2 to both terms inside the parentheses: 1.2 ∗ 6.3 and 1.2 ∗ − 7 x . We have 1.2 × 6.3 = 7.56 and 1.2 × − 7 x = − 8.4 x . So the equation becomes 3.5 + 7.56 − 8.4 x = 9.38 .
Combine constants Next, we combine the constant terms on the left side of the equation: 3.5 + 7.56 = 11.06 . The equation is now 11.06 − 8.4 x = 9.38 .
Isolate x term Now, we isolate the term with x by subtracting the constant term from both sides of the equation: 11.06 − 8.4 x − 11.06 = 9.38 − 11.06 , which simplifies to − 8.4 x = − 1.68 .
Solve for x Finally, we solve for x by dividing both sides of the equation by the coefficient of x : − 8.4 − 8.4 x = − 8.4 − 1.68 , which gives x = 0.2 .
Identify correct steps Based on these steps, the correct options are:
Distribute 1.2 to 6.3 and − 7 x .
Combine 3.5 and 7.56.
Subtract 11.06 from both sides.
Examples
Solving linear equations like this is crucial in many real-world scenarios. For instance, imagine you're managing a budget for a project. You have a fixed amount to spend, and certain costs are dependent on a variable, like the number of hours worked. By setting up and solving a linear equation, you can determine the maximum number of hours you can afford without exceeding your budget. This kind of problem-solving is fundamental in finance, engineering, and many other fields where resource allocation is critical. For example, if the equation represents the cost of materials and labor, solving for the variable (e.g., number of units produced) helps in determining production capacity within a set budget.
