Drag each tile to the correct box.
Arrange the equations in order from least to greatest based on their solution.
Equation A: [tex]$\quad 5(x-6)+3 x=\frac{3}{4}(2 x-8)$[/tex]
Equation B: [tex]$\quad 2.7(5.1 x+4.9)=3.2 x+28.9$[/tex]
Equation C: [tex]$\quad 5(11 x-18)=3(2 x+7)$[/tex]
[tex]$\ \textless \ $[/tex] [tex]$\square$[/tex] [tex]$\ \textless \ $[/tex] [tex]$\square$[/tex]
Answer (2)
Solve Equation A: 5 ( x − 6 ) + 3 x = 4 3 ( 2 x − 8 ) which simplifies to x = 13 48 ≈ 3.69 .
Solve Equation B: 2.7 ( 5.1 x + 4.9 ) = 3.2 x + 28.9 which simplifies to x = 10.57 15.67 ≈ 1.48 .
Solve Equation C: 5 ( 11 x − 18 ) = 3 ( 2 x + 7 ) which simplifies to x = 49 111 ≈ 2.27 .
Arrange the solutions in increasing order: Equation B < Equation C < Equation A, thus 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 = 4 6 x − 4 24 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 = 24 Multiply both sides by 13 2 :
x = 24 × 13 2 = 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: 13.77 x − 3.2 x + 13.23 = 28.9 10.57 x = 28.9 − 13.23 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: 55 x − 6 x − 90 = 21 49 x = 21 + 90 49 x = 111 Divide both sides by 49: x = 49 111 ≈ 2.27
Comparing Solutions Now we compare the solutions: Equation A has x ≈ 3.69 , Equation B has x ≈ 1.48 , and Equation C has x ≈ 2.27 . Arranging these in increasing order, we have 1.48 < 2.27 < 3.69 . Therefore, the order of the equations from least to greatest based on their solution is Equation B < Equation C < Equation A.
Final Answer The equations arranged from least to greatest based on their solution are: Equation B < Equation C < Equation A.
Examples
Understanding how to solve and compare linear equations is crucial in many real-world scenarios. For instance, imagine you're comparing different phone plans. Each plan has a fixed monthly fee and a per-minute charge. By setting up linear equations for each plan and solving for the number of minutes you'd need to use for the plans to cost the same, you can determine which plan is more economical for your usage. Similarly, in business, comparing costs and revenues using linear equations helps in making informed decisions about pricing and production levels.
After solving the equations, we find that Equation B has the least solution (.48), followed by Equation C (.27), and Equation A has the greatest solution (.69). Therefore, the order from least to greatest is B < C < A.
;
