IDNLearn.com provides a collaborative environment for finding and sharing knowledge. Receive prompt and accurate responses to your questions from our community of knowledgeable professionals ready to assist you at any time.
Sagot :
Let's analyze each of the given equations step-by-step to classify them according to the type of solution they have. We will determine if an equation has no solution, one solution, or infinitely many solutions.
### Equation 1: [tex]\(\frac{1}{2} y + 3.2 y = 20\)[/tex]
First, let's combine like terms:
[tex]\[ \left(\frac{1}{2} + 3.2\right) y = 20 \][/tex]
This simplifies to:
[tex]\[ 3.7 y = 20 \][/tex]
To solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{20}{3.7} \][/tex]
This equation has one solution.
### Equation 2: [tex]\(\frac{15}{2} + 2 z - \frac{1}{4} = 4 z + \frac{20}{4} - 2 z + 3 z + 2.5 \)[/tex]
Simplify both sides of the equation:
[tex]\[ \frac{15}{2} + 2 z - \frac{1}{4} = 4 z + 5 - 2 z + 3 z + 2.5 \][/tex]
By simplifying the constants and combining the terms involving [tex]\( z \)[/tex]:
[tex]\[ \frac{30}{4} + 8z/4 - 1/4 = 4z + 5 - 2z + 3z + 2.5 \][/tex]
[tex]\[ 7.25 + 2z = 5z + 7.5 \][/tex]
Rearrange to solve for [tex]\( z \)[/tex]:
[tex]\[ 7.25 - 7.5 = 5z - 2z \][/tex]
[tex]\[ -0.25 = 3z \][/tex]
[tex]\[ z = -\frac{0.25}{3} \][/tex]
This equation has one solution.
### Equation 3: [tex]\(1.1 + \frac{3}{4} x + 2 = 3.1 + \frac{3}{4} x\)[/tex]
Simplify and combine like terms on both sides:
[tex]\[ \left(1.1 + 2\right) + \frac{3}{4} x = 3.1 + \frac{3}{4} x \][/tex]
[tex]\[ 3.1 + \frac{3}{4} x = 3.1 + \frac{3}{4} x \][/tex]
This holds true for all [tex]\( x \)[/tex].
This equation has infinitely many solutions.
### Equation 4: [tex]\(4.5 r = 3.2 + 4.5 r\)[/tex]
To solve this equation, isolate the term involving [tex]\( r \)[/tex] on one side:
[tex]\[ 4.5 r - 4.5 r = 3.2 \][/tex]
[tex]\[ 0 = 3.2 \][/tex]
This is a contradiction.
This equation has no solution.
### Equation 5: [tex]\(2 x + 4 = 3 x + \frac{1}{2}\)[/tex]
Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[ 2 x + 4 = 3 x + \frac{1}{2} \][/tex]
[tex]\[ 4 - \frac{1}{2} = 3 x - 2 x \][/tex]
[tex]\[ \frac{7.5}{4} = x \][/tex]
This equation has one solution.
### Classification Summary:
- No Solution: [tex]\(4.5 r = 3.2 + 4.5 r\)[/tex]
- One Solution:
- [tex]\(\frac{1}{2} y + 3.2 y = 20\)[/tex]
- [tex]\(\frac{15}{2} + 2 z - \frac{1}{4} = 4 z + \frac{20}{4} - 2 z + 3 z + 2.5 = 3.2 + 3 z \)[/tex]
- [tex]\(2 x + 4 = 3 x + \frac{1}{2} \)[/tex]
- Infinitely Many Solutions:
- [tex]\(1.1 + \frac{3}{4} x + 2 = 3.1 + \frac{3}{4} x\)[/tex]
Hence, the completed table is:
[tex]\[ \begin{tabular}{|l|l|} \hline No Solution & \(4.5 r = 3.2 + 4.5 r\) \\ \hline One Solution & \begin{tabular}{l} \(\frac{1}{2} y + 3.2 y = 20\) \\ \(\frac{15}{2} + 2 z - \frac{1}{4} = 4 z + \frac{20}{4} - 2 z + 3 z + 2.5 = 3.2 + 3 z\) \\ \(2 x + 4 = 3 x + \frac{1}{2}\) \\ \end{tabular} \\ \hline Infinitely Many Solutions & \(1.1 + \frac{3}{4} x + 2 = 3.1 + \frac{3}{4} x\) \\ \hline \end{tabular} \][/tex]
### Equation 1: [tex]\(\frac{1}{2} y + 3.2 y = 20\)[/tex]
First, let's combine like terms:
[tex]\[ \left(\frac{1}{2} + 3.2\right) y = 20 \][/tex]
This simplifies to:
[tex]\[ 3.7 y = 20 \][/tex]
To solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{20}{3.7} \][/tex]
This equation has one solution.
### Equation 2: [tex]\(\frac{15}{2} + 2 z - \frac{1}{4} = 4 z + \frac{20}{4} - 2 z + 3 z + 2.5 \)[/tex]
Simplify both sides of the equation:
[tex]\[ \frac{15}{2} + 2 z - \frac{1}{4} = 4 z + 5 - 2 z + 3 z + 2.5 \][/tex]
By simplifying the constants and combining the terms involving [tex]\( z \)[/tex]:
[tex]\[ \frac{30}{4} + 8z/4 - 1/4 = 4z + 5 - 2z + 3z + 2.5 \][/tex]
[tex]\[ 7.25 + 2z = 5z + 7.5 \][/tex]
Rearrange to solve for [tex]\( z \)[/tex]:
[tex]\[ 7.25 - 7.5 = 5z - 2z \][/tex]
[tex]\[ -0.25 = 3z \][/tex]
[tex]\[ z = -\frac{0.25}{3} \][/tex]
This equation has one solution.
### Equation 3: [tex]\(1.1 + \frac{3}{4} x + 2 = 3.1 + \frac{3}{4} x\)[/tex]
Simplify and combine like terms on both sides:
[tex]\[ \left(1.1 + 2\right) + \frac{3}{4} x = 3.1 + \frac{3}{4} x \][/tex]
[tex]\[ 3.1 + \frac{3}{4} x = 3.1 + \frac{3}{4} x \][/tex]
This holds true for all [tex]\( x \)[/tex].
This equation has infinitely many solutions.
### Equation 4: [tex]\(4.5 r = 3.2 + 4.5 r\)[/tex]
To solve this equation, isolate the term involving [tex]\( r \)[/tex] on one side:
[tex]\[ 4.5 r - 4.5 r = 3.2 \][/tex]
[tex]\[ 0 = 3.2 \][/tex]
This is a contradiction.
This equation has no solution.
### Equation 5: [tex]\(2 x + 4 = 3 x + \frac{1}{2}\)[/tex]
Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[ 2 x + 4 = 3 x + \frac{1}{2} \][/tex]
[tex]\[ 4 - \frac{1}{2} = 3 x - 2 x \][/tex]
[tex]\[ \frac{7.5}{4} = x \][/tex]
This equation has one solution.
### Classification Summary:
- No Solution: [tex]\(4.5 r = 3.2 + 4.5 r\)[/tex]
- One Solution:
- [tex]\(\frac{1}{2} y + 3.2 y = 20\)[/tex]
- [tex]\(\frac{15}{2} + 2 z - \frac{1}{4} = 4 z + \frac{20}{4} - 2 z + 3 z + 2.5 = 3.2 + 3 z \)[/tex]
- [tex]\(2 x + 4 = 3 x + \frac{1}{2} \)[/tex]
- Infinitely Many Solutions:
- [tex]\(1.1 + \frac{3}{4} x + 2 = 3.1 + \frac{3}{4} x\)[/tex]
Hence, the completed table is:
[tex]\[ \begin{tabular}{|l|l|} \hline No Solution & \(4.5 r = 3.2 + 4.5 r\) \\ \hline One Solution & \begin{tabular}{l} \(\frac{1}{2} y + 3.2 y = 20\) \\ \(\frac{15}{2} + 2 z - \frac{1}{4} = 4 z + \frac{20}{4} - 2 z + 3 z + 2.5 = 3.2 + 3 z\) \\ \(2 x + 4 = 3 x + \frac{1}{2}\) \\ \end{tabular} \\ \hline Infinitely Many Solutions & \(1.1 + \frac{3}{4} x + 2 = 3.1 + \frac{3}{4} x\) \\ \hline \end{tabular} \][/tex]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.
