(1) Simplify $3-(8 r-3)+9=\frac{6}{5}(3 r-2)$ and represent your answer on a number line. (2) Evaluate $\frac{5 k-3}{9}=\frac{3}{7}(3-2 k)$ and state the Solution Set.
Answer (2)
The solution to the first equation is r = 1.5 , and the corresponding point can be represented on a number line. The solution for the second equation is k = 89 102 , with the solution set being { \frac{102}{89} }.
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Simplify the first equation 3 − ( 8 r − 3 ) + 9 = 5 6 ( 3 r − 2 ) to find r .
Solve for r and get r = 2 3 = 1.5 .
Evaluate the second equation 9 5 k − 3 = 7 3 ( 3 − 2 k ) to find k .
Solve for k and get k = 89 102 .
The solution for the first equation is r = 1.5 , and the solution set for the second equation is { 89 102 } .
Explanation
Problem Analysis We are given two equations to solve. The first equation is 3 − ( 8 r − 3 ) + 9 = 5 6 ( 3 r − 2 ) , and we need to simplify it and represent the solution on a number line. The second equation is 9 5 k − 3 = 7 3 ( 3 − 2 k ) , and we need to evaluate it and state the solution set.
Simplifying the First Equation First, let's simplify the first equation: 3 − ( 8 r − 3 ) + 9 = 5 6 ( 3 r − 2 ) .
Distribute the negative sign: 3 − 8 r + 3 + 9 = 5 6 ( 3 r − 2 ) .
Combine like terms on the left side: 15 − 8 r = 5 6 ( 3 r − 2 ) .
Distribute 5 6 on the right side: 15 − 8 r = 5 18 r − 5 12 .
Add 8 r to both sides: 15 = 5 18 r + 8 r − 5 12 .
Convert 8 r to a fraction with a denominator of 5: 15 = 5 18 r + 5 40 r − 5 12 .
Combine the r terms: 15 = 5 58 r − 5 12 .
Add 5 12 to both sides: 15 + 5 12 = 5 58 r .
Convert 15 to a fraction with a denominator of 5: 5 75 + 5 12 = 5 58 r .
Combine the fractions on the left side: 5 87 = 5 58 r .
Multiply both sides by 58 5 : r = 5 87 × 58 5 .
Simplify: r = 58 87 = 2 × 29 3 × 29 = 2 3 = 1.5 .
Representing the Solution on a Number Line Now, let's represent the solution r = 1.5 on a number line. We simply draw a number line and mark the point 1.5 on it.
Evaluating the Second Equation Next, let's evaluate the second equation: 9 5 k − 3 = 7 3 ( 3 − 2 k ) .
Multiply both sides by 9 and 7 to eliminate the fractions: 7 ( 5 k − 3 ) = 9 × 3 ( 3 − 2 k ) .
Simplify: 35 k − 21 = 27 ( 3 − 2 k ) .
Distribute the 27: 35 k − 21 = 81 − 54 k .
Add 54 k to both sides: 35 k + 54 k − 21 = 81 .
Combine the k terms: 89 k − 21 = 81 .
Add 21 to both sides: 89 k = 81 + 21 .
Simplify: 89 k = 102 .
Divide both sides by 89: k = 89 102 ≈ 1.146 .
Stating the Solution Set The solution set for the second equation is { 89 102 } .
Final Answer Therefore, the solution to the first equation is r = 2 3 = 1.5 , and the solution set for the second equation is { 89 102 } .
Examples
Imagine you are balancing a seesaw. The first equation is like finding the exact point to place yourself to balance it perfectly. The second equation is like adjusting the weights on each side to achieve equilibrium. Solving these equations helps in real-world scenarios like balancing budgets, mixing ingredients in recipes, or calculating distances and speeds.
