Drag each tile to the correct box.
Arrange the equations in order from least to greatest based on their solution.
Equation A: [tex]5(x-6)+3 x=\frac{3}{4}(2 x-8)[/tex]
Equation B: [tex]2.7(5.1 x+4.9)=3.2 x+28.9[/tex]
Equation C: [tex]5(11 x-18)=3(2 x+7)[/tex]
Equation A < $\square$ < $\square$
Equation B
Equation C
Answer (2)
Solve Equation A: 5 ( x − 6 ) + 3 x = 4 3 ( 2 x − 8 ) which simplifies to x ≈ 3.69 .
Solve Equation B: 2.7 ( 5.1 x + 4.9 ) = 3.2 x + 28.9 which simplifies to x ≈ 1.48 .
Solve Equation C: 5 ( 11 x − 18 ) = 3 ( 2 x + 7 ) which simplifies to x ≈ 2.27 .
Arrange the equations based on their solutions from least to greatest: B < C < A .
Explanation
Problem Analysis We are given three equations and asked to arrange them in order from least to greatest based on their solutions. To do this, we must first solve each equation for x .
Solving Equation A Equation A: 5 ( x − 6 ) + 3 x = 4 3 ( 2 x − 8 )
First, distribute the constants on both sides of the equation: 5 x − 30 + 3 x = 2 3 x − 6 Combine like terms: 8 x − 30 = 2 3 x − 6 Subtract 2 3 x from both sides: 8 x − 2 3 x − 30 = − 6 2 16 x − 2 3 x − 30 = − 6 2 13 x − 30 = − 6 Add 30 to both sides: 2 13 x = 24 Multiply both sides by 13 2 :
x = 13 48 ≈ 3.69
Solving Equation B Equation B: 2.7 ( 5.1 x + 4.9 ) = 3.2 x + 28.9 Distribute the constant on the left side: 13.77 x + 13.23 = 3.2 x + 28.9 Subtract 3.2 x from both sides: 10.57 x + 13.23 = 28.9 Subtract 13.23 from both sides: 10.57 x = 15.67 Divide both sides by 10.57: x = 10.57 15.67 ≈ 1.48
Solving Equation C Equation C: 5 ( 11 x − 18 ) = 3 ( 2 x + 7 ) Distribute the constants on both sides: 55 x − 90 = 6 x + 21 Subtract 6 x from both sides: 49 x − 90 = 21 Add 90 to both sides: 49 x = 111 Divide both sides by 49: x = 49 111 ≈ 2.27
Comparing Solutions and Ordering Comparing the solutions, we have: Equation A: x ≈ 3.69 Equation B: x ≈ 1.48 Equation C: x ≈ 2.27 Arranging these in ascending order, we get: Equation B < Equation C < Equation A
Examples
Understanding how to solve and compare linear equations is crucial in many real-world scenarios, such as comparing costs, analyzing data, and making informed decisions. For instance, imagine you are comparing different phone plans. Each plan has a different monthly fee and a different cost per gigabyte of data. By setting up linear equations for each plan and solving for the total cost based on your data usage, you can determine which plan is the most cost-effective for your needs. This involves solving linear equations and comparing the results, just like in the problem we solved.
To arrange the equations by their solutions, we found that Equation B has the smallest solution (approximately 1.48), followed by Equation C (approximately 2.27), and lastly Equation A (approximately 3.69). Thus, the order is Equation B < Equation C < Equation A.
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