Certainly! Let's analyze each of the given equations step-by-step to determine which of them describe straight lines.
1. Equation: [tex]\(2y - 9 = 4x\)[/tex]
- To check if this equation describes a straight line, we need to rewrite it in the form [tex]\(y = mx + c\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(c\)[/tex] is the y-intercept.
- Let's isolate [tex]\(y\)[/tex]:
[tex]\[
2y = 4x + 9
\][/tex]
[tex]\[
y = 2x + \frac{9}{2}
\][/tex]
- Since this is in the form [tex]\(y = mx + c\)[/tex], it describes a straight line.
2. Equation: [tex]\(3y = 6x\)[/tex]
- Again, we check if it can be written in the form [tex]\(y = mx + c\)[/tex].
- Isolate [tex]\(y\)[/tex]:
[tex]\[
y = \frac{6}{3}x
\][/tex]
[tex]\[
y = 2x
\][/tex]
- This is already in the form [tex]\(y = mx + c\)[/tex] with a slope of [tex]\(2\)[/tex] and y-intercept [tex]\(0\)[/tex]. Therefore, it describes a straight line.
3. Equation: [tex]\(y = 5x^2 + 1\)[/tex]
- This equation includes an [tex]\(x^2\)[/tex] term, which means it is quadratic, not linear.
- Since a quadratic equation does not fit the form [tex]\(y = mx + c\)[/tex], it does not describe a straight line.
4. Equation: [tex]\(y = 7\)[/tex]
- This is already in the form [tex]\(y = c\)[/tex], which represents a horizontal line where [tex]\(y\)[/tex] is always [tex]\(7\)[/tex], regardless of the value of [tex]\(x\)[/tex].
- This is indeed a straight line.
5. Equation: [tex]\(y + 4 = \frac{3}{x}\)[/tex]
- To check if this equation is linear, try to isolate [tex]\(y\)[/tex]:
[tex]\[
y = \frac{3}{x} - 4
\][/tex]
- The term [tex]\(\frac{3}{x}\)[/tex] is not linear because it involves the variable [tex]\(x\)[/tex] in the denominator.
- Therefore, this does not fit the form [tex]\(y = mx + c\)[/tex] and does not describe a straight line.
In summary, the three equations that describe straight lines are:
[tex]\[2y - 9 = 4x, \quad 3y = 6x, \quad y = 7\][/tex]
