To determine which of the given equations represent linear relationships, we need to examine each equation and see if it can be written in the form of a linear equation, which is typically given by \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants, and \(x\) and \(y\) are variables.
Let's analyze each equation one by one:
1. Equation: \(5 + 2y = 13\)
- Rearrange to isolate \(y\): \(2y = 13 - 5 = 8\)
- Solve for \(y\): \(y = \frac{8}{2} = 4\)
- This can be written as \(y = 4\), which is a linear equation.
2. Equation: \(y = \frac{1}{2} x^2 + 7\)
- This equation includes the term \(x^2\), which is a quadratic term.
- Since it involves \(x^2\), it is not a linear equation.
3. Equation: \(y - 5 = 2(x - 1)\)
- Distribute and rearrange: \(y - 5 = 2x - 2\)
- Solve for \(y\): \(y = 2x - 2 + 5 = 2x + 3\)
- This can be written as \(y = 2x + 3\), which is a linear equation.
4. Equation: \(\frac{y}{2} = x + 7\)
- Multiply both sides by 2 to isolate \(y\): \(y = 2(x + 7) = 2x + 14\)
- This can be written as \(y = 2x + 14\), which is a linear equation.
5. Equation: \(x = -4\)
- This represents a vertical line where \(x\) is always \(-4\), independent of \(y\).
- In the \(xy\)-plane, this is still a linear relationship, though it's a special case where it doesn't explicitly involve \(y\), but it is a valid linear equation.
Summary of the Analysis:
- Equation 1 \(5 + 2y = 13\) \( \Rightarrow \) Linear
- Equation 2 \(y = \frac{1}{2} x^2 + 7\) \( \Rightarrow \) Not Linear
- Equation 3 \(y - 5 = 2(x - 1)\) \( \Rightarrow \) Linear
- Equation 4 \(\frac{y}{2} = x + 7\) \( \Rightarrow \) Linear
- Equation 5 \(x = -4\) \( \Rightarrow \) Linear
Therefore, the equations that represent linear relationships are:
- \(5 + 2y = 13\)
- \(y - 5 = 2(x - 1)\)
- \(\frac{y}{2} = x + 7\)
- [tex]\(x = -4\)[/tex]
