9 answers
In the steps you provided, we can identify the justification for moving from the equation \( \frac{3z}{2} - 5 = 10 \) to \( \frac{3z}{2} - 5 + 5 = 10 + 5 \).
The step you are looking at involves adding 5 to both sides of the equation. This is justified by the Addition Property of Equality, which states that if you add the same amount to both sides of an equation, the sides remain equal.
Therefore, the correct response is:
Addition Property of Equality.
To solve the equation
\[ \frac{2f}{5} - \frac{8}{5} + 2f = 8, \]
first, we'll combine like terms.
\[ 2f = \frac{10f}{5}. \]
Now rewrite the equation:
\[ \frac{2f}{5} + \frac{10f}{5} - \frac{8}{5} = 8. \]
Now combine the terms on the left:
\[ \frac{2f + 10f - 8}{5} = 8. \]
This simplifies to:
\[ \frac{12f - 8}{5} = 8. \]
\[ 12f - 8 = 40. \]
\[ 12f = 40 + 8, \]
which gives:
\[ 12f = 48. \]
\[ f = \frac{48}{12} = 4. \]
Thus, the solution to the equation is:
4.
To determine how many solutions the equation
\[ \frac{2}{3}(6m + 9) - 6 = 4m \]
has, let's solve it step by step.
Distribute \(\frac{2}{3}\):
\[ \frac{2}{3} \times 6m + \frac{2}{3} \times 9 - 6 = 4m. \]
This simplifies to:
\[ 4m + 6 - 6 = 4m. \]
Simplify the equation:
\[ 4m = 4m. \]
This equation is true for any value of \(m\) since the left side is equal to the right side. Therefore, there are infinitely many solutions.
Thus, the correct response is:
infinite/many solutions.
To identify the pair of linear equations that have the same solution set, we can solve each pair of equations and see if they yield the same solution.
Pair 1:
For the first equation: \[ m + 1 = 10 \implies m = 9. \]
For the second equation, rearranging gives: \[ 6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7}. \]
So, they do not have the same solution.
Pair 2:
For the first equation: \[ 4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2}. \]
For the second equation: \[ 8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5}. \]
So, these also do not have the same solution.
Pair 3:
For the first equation: \[ 6g + 10 - 2g = 8 - 2g \implies 6g + 10 = 8 - 2g \] Rearranging gives: \[ 6g + 2g = 8 - 10 \implies 8g = -2 \implies g = -\frac{1}{4}. \]
For the second equation: \[ -36 = 6h \implies h = -6. \]
So, they do not have the same solution.
Pair 4:
For the first equation: \[ \frac{k}{2} = 4 - 5 \implies \frac{k}{2} = -1 \implies k = -2. \]
For the second equation: \[ \frac{4j - 14}{3} = 3 - 5 \implies \frac{4j - 14}{3} = -2. \] Cross-multiplying gives: \[ 4j - 14 = -6 \implies 4j = 8 \implies j = 2. \]
So, these also do not have the same solution.
After analyzing all pairs, it appears that none of the pairs provided have the same solution set.
However, rereading the operations can offer insights; pairs are found in context easily.
Given the calculations and conclusions above, pick pairs that, through similar operational setups outlined in expressions, could have mirrored outputs. Thus, evaluate particularly foundational approaches. None were aptly matched from provided outputs.
If one of these ought to mirror well among structured equations, we might need to re-evaluate from ground stage ideally to construct equivalency akin.
Conclusion: Determination of pairs through evaluation led to identification errors in simplification contexts. Output observations contextual to linear equations reflect upon analytical assessments to achieve accurate articulations for correlated outputs, deciphering verification through mathematical scrutiny through processes seen to enhance clarity.
To identify the pair of linear equations that have the same solution set, let's solve each pair of equations step-by-step.
Pair 1:
First equation: \[ 3(m + 1) = 10 \implies m + 1 = \frac{10}{3} \implies m = \frac{10}{3} - 1 = \frac{10}{3} - \frac{3}{3} = \frac{7}{3}. \]
Second equation: \[ 6n + 3 = 6 - n \implies 6n + n = 6 - 3 \implies 7n = 3 \implies n = \frac{3}{7}. \]
Result: \( m = \frac{7}{3}, n = \frac{3}{7} \) (Not the same solution set)
Pair 2:
First equation: \[ 4k - 5 = 2k \implies 4k - 2k = 5 \implies 2k = 5 \implies k = \frac{5}{2}. \]
Second equation: \[ 8j - 2 = 3j \implies 8j - 3j = 2 \implies 5j = 2 \implies j = \frac{2}{5}. \]
Result: \( k = \frac{5}{2}, j = \frac{2}{5} \) (Not the same solution set)
Pair 3:
First equation: \[ 2(3g + 5) - 2g = 2(4 - g) \] Expanding this gives: \[ 6g + 10 - 2g = 8 - 2g \implies 6g + 10 = 8 - 2g. \] Rearranging: \[ 6g + 2g = 8 - 10 \implies 8g = -2 \implies g = -\frac{1}{4}. \]
Second equation: \[ -\frac{36}{h} = 6 \implies 36 = -6h \implies h = -6. \]
Result: \( g = -\frac{1}{4}, h = -6 \) (Not the same solution set)
Pair 4:
First equation: \[ 4 = \frac{k}{2} + 5 \implies \frac{k}{2} = 4 - 5 \implies \frac{k}{2} = -1 \implies k = -2. \]
Second equation: \[ \frac{4j - 14}{3} + 5 = 3 \implies \frac{4j - 14}{3} = 3 - 5 \implies \frac{4j - 14}{3} = -2. \] Cross multiplying gives: \[ 4j - 14 = -6 \implies 4j = 8 \implies j = 2. \]
Result: \( k = -2, j = 2 \) (Not the same solution set)
After analyzing all pairs, it seems they all yield different solutions.
However, given the structure of the equations, it is possible we misinterpret the context or approach mathematically on outputs.
To clarify, none of the stated pairs have the same solution upon validation through steps provided. Thus, we might need to reassess to uncover alternative connections or dual-linear relationships yielding mirrored equivalencies not captured succinctly present.
For any correction or oversight analysis - if clarity yields symmetry or coherence differentials – direct insight would be beneficial.
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