Drag each tile to the correct location on the table. Each tile can be used more than once, but not all tiles will be used.
Choose the justification for each step in the solution to the given equation.
simplification
addition property of equality
subtraction property of equality
multiplication property of equality
division property of equality
| Step | Justification |
| :----------------------------------------------------------------- | :------------ |
| [tex]\frac{17}{3}-\frac{3}{4} x=\frac{1}{2} x+5[/tex] | given |
| [tex]\frac{17}{3}-\frac{3}{4} x-\frac{17}{3}=\frac{1}{2} x+5-\frac{17}{3}[/tex] | |
| [tex]-\frac{3}{4} x=\frac{1}{2} x-\frac{2}{3}[/tex] | |
| [tex]-\frac{3}{4} x-\frac{1}{2} x=\frac{1}{2} x-\frac{2}{3}-\frac{1}{2} x[/tex] | |
| [tex]-\frac{5}{4} x=-\frac{2}{3}[/tex] | |
| [tex]-\frac{5}{4} x \cdot-\frac{4}{5}=-\frac{2}{3} \cdot-\frac{4}{5}[/tex] | |
| [tex]x=\frac{8}{15}[/tex] | |
Answer (1)
Subtract 3 17 from both sides: subtraction property of equality.
Simplify the equation: simplification.
Subtract 2 1 x from both sides: subtraction property of equality.
Simplify again: simplification.
Multiply both sides by − 5 4 : multiplication property of equality.
Simplify to find the value of x: simplification.
The solution is x = 15 8 .
Explanation
Understanding the Problem We are given an equation and a step-by-step solution. Our task is to provide the correct justification for each step in the solution.
Given Equation The first step is 3 17 − 4 3 x = 2 1 x + 5 , which is given.
Subtraction Property of Equality The second step is 3 17 − 4 3 x − 3 17 = 2 1 x + 5 − 3 17 . We obtain this by subtracting 3 17 from both sides of the equation. This is the subtraction property of equality.
Simplification The third step is − 4 3 x = 2 1 x − 3 2 . This step simplifies the previous equation. On the left side, 3 17 − 4 3 x − 3 17 = − 4 3 x . On the right side, we have 5 − 3 17 = 3 15 − 3 17 = − 3 2 . Thus, the justification is simplification.
Subtraction Property of Equality The fourth step is − 4 3 x − 2 1 x = 2 1 x − 3 2 − 2 1 x . We obtain this by subtracting 2 1 x from both sides of the equation. This is the subtraction property of equality.
Simplification The fifth step is − 4 5 x = − 3 2 . This step simplifies the previous equation. On the left side, we have − 4 3 x − 2 1 x = − 4 3 x − 4 2 x = − 4 5 x . Thus, the justification is simplification.
Multiplication Property of Equality The sixth step is − 4 5 x ⋅ − 5 4 = − 3 2 ⋅ − 5 4 . We obtain this by multiplying both sides of the equation by − 5 4 . This is the multiplication property of equality.
Simplification The seventh step is x = 15 8 . This step simplifies the previous equation. On the right side, we have − 3 2 ⋅ − 5 4 = 15 8 . Thus, the justification is simplification.
Examples
Understanding how to solve equations step-by-step is crucial in many real-world scenarios. For example, imagine you're trying to determine how many hours you need to work to save up for a new bicycle. By setting up an equation that represents your savings and expenses, you can use these algebraic properties to isolate the variable (number of hours) and find the solution. Each step, like adding or subtracting expenses, directly applies these properties to reach your goal. This process is also fundamental in fields like engineering, finance, and computer science, where problem-solving often involves manipulating equations to find desired outcomes.
