- Equation 1 simplifies to \(-6x + 31 = -6x - 25\), which has no solutions because \(-6 = -6\) but \(31
eq -25\).
- Equation 2 simplifies to \(8x - 12 = -8x - 12\), which has one solution because \(8
eq -8\).
- Equation 3 simplifies to \(4m - 36 = 4m - 38\), which has no solutions because \(4 = 4\) but \(-36
eq -38\).
- Final results: Equation 1: \(\boxed{\text{no solutions}}\); Equation 2: \(\boxed{\text{one solution}}\); Equation 3: \(\boxed{\text{no solutions}}\).
Explanation
- Analysis of the Linear Equations
To determine the number of solutions for each given linear equation, we simplify each equation into the standard form \(Ax + B = Cx + D\). A linear equation of this form has:
- Exactly one solution if \(A
eq C\).
- No solutions if \(A = C\) and \(B
eq D\).
- Infinitely many solutions if \(A = C\) and \(B = D\).
We will analyze each equation step by step.
- Solving Equation 1: \(-3(2x - 9) + 4 = -6x - 25\)
Consider the first equation:
\[-3(2x - 9) + 4 = -6x - 25\]
First, distribute the \(-3\) on the left side:
\[-6x + 27 + 4 = -6x - 25\]
Combine like terms:
\[-6x + 31 = -6x - 25\]
Comparing this to the form \(Ax + B = Cx + D\), we have \(A = -6\), \(B = 31\), \(C = -6\), and \(D = -25\).
Since \(A = C\) but \(B
eq D\) (\(31
eq -25\)), this equation has no solutions.
- Solving Equation 2: \(8x - 12 = -8x - 12\)
Consider the second equation:
\[8x - 12 = -8x - 12\]
Comparing this directly to the form \(Ax + B = Cx + D\), we have \(A = 8\), \(B = -12\), \(C = -8\), and \(D = -12\).
Since \(A
eq C\) (\(8
eq -8\)), this equation has exactly one solution.
- Solving Equation 3: \(4(m - 9) = 2(2m + 1) - 40\)
Consider the third equation:
\[4(m - 9) = 2(2m + 1) - 40\]
First, expand both sides:
\[4m - 36 = 4m + 2 - 40\]
Combine like terms on the right side:
\[4m - 36 = 4m - 38\]
Comparing this to the form \(Am + B = Cm + D\), we have \(A = 4\), \(B = -36\), \(C = 4\), and \(D = -38\).
Since \(A = C\) but \(B
eq D\) (\(-36
eq -38\)), this equation has no solutions.
- Conclusion and Categorization
In summary:
- Equation 1 (\(-3(2x - 9) + 4 = -6x - 25\)) results in no solutions.
- Equation 2 (\(8x - 12 = -8x - 12\)) results in one solution.
- Equation 3 (\(4(m - 9) = 2(2m + 1) - 40\)) results in no solutions.
According to the provided key:
- Equation 1 corresponds to the "no" box.
- Equation 2 corresponds to the "one" box.
- Equation 3 corresponds to the "no" box.
Thus, we categorize these equations as follows:
Equation 1: \(\boxed{\text{no solutions}}\), Equation 2: \(\boxed{\text{one solution}}\), Equation 3: \(\boxed{\text{no solutions}}\).
Examples
Understanding the nature of linear equations is crucial in fields like economics and engineering. For example, when solving a system to find the equilibrium point between supply and demand, finding no solution might indicate that the market conditions are impossible, a single solution shows a unique point of balance, and infinite solutions suggest that the supply and demand curves are identical, implying constant equilibrium across a range of prices. Recognizing these patterns allows us to make informed decisions in real-world modeling.