Let's analyze each equation to see if it represents a linear relationship.
1. Equation: [tex]\( 5 + 2y = 13 \)[/tex]
To determine if this equation represents a linear relationship, we'll solve for [tex]\( y \)[/tex]:
[tex]\[
5 + 2y = 13
\][/tex]
Subtract 5 from both sides:
[tex]\[
2y = 8
\][/tex]
Divide by 2:
[tex]\[
y = 4
\][/tex]
This is a linear equation because it can be written in the form [tex]\( y = mx + b \)[/tex] where [tex]\( m = 0 \)[/tex] and [tex]\( b = 4 \)[/tex]. Therefore, this is a linear equation.
2. Equation: [tex]\( y = \frac{1}{2} x^2 + 7 \)[/tex]
This equation explicitly includes an [tex]\( x^2 \)[/tex] term, which means it is a quadratic equation, not linear. Linear equations can only have variables to the first power. Therefore, this is not a linear equation.
3. Equation: [tex]\( y - 5 = 2(x - 1) \)[/tex]
To determine if this equation represents a linear relationship, we'll rearrange it into the form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y - 5 = 2(x - 1)
\][/tex]
Distribute the 2 on the right side:
[tex]\[
y - 5 = 2x - 2
\][/tex]
Add 5 to both sides:
[tex]\[
y = 2x + 3
\][/tex]
This is a linear equation because it can be written in the form [tex]\( y = mx + b \)[/tex] where [tex]\( m = 2 \)[/tex] and [tex]\( b = 3 \)[/tex]. Therefore, this is a linear equation.
4. Equation: [tex]\( \frac{y}{2} = x + 7 \)[/tex]
To determine if this equation represents a linear relationship, we'll solve for [tex]\( y \)[/tex]:
[tex]\[
\frac{y}{2} = x + 7
\][/tex]
Multiply both sides by 2:
[tex]\[
y = 2x + 14
\][/tex]
This is a linear equation because it can be written in the form [tex]\( y = mx + b \)[/tex] where [tex]\( m = 2 \)[/tex] and [tex]\( b = 14 \)[/tex]. Therefore, this is a linear equation.
5. Equation: [tex]\( x = -4 \)[/tex]
This equation represents a vertical line on the coordinate plane, which means that [tex]\( x \)[/tex] remains constant and there is no dependence on [tex]\( y \)[/tex]. Such an equation is still considered linear, though it doesn't fit the [tex]\( y = mx + b \)[/tex] format because it is essentially of the form [tex]\( x = c \)[/tex] where [tex]\( c \)[/tex] represents a constant. Therefore, this is a linear equation.
Based on our analysis:
- Equation [tex]\( 5 + 2y = 13 \)[/tex] is not linear.
- Equation [tex]\( y = \frac{1}{2} x^2 + 7 \)[/tex] is not linear.
- Equation [tex]\( y - 5 = 2(x - 1) \)[/tex] is linear.
- Equation [tex]\( \frac{ y }{2}=x+7 \)[/tex] is linear.
- Equation [tex]\( x = -4 \)[/tex] is linear.
Thus, the equations that represent linear relationships are:
[tex]\[ y - 5 = 2(x - 1) \][/tex]
[tex]\[ \frac{ y }{2}=x+7 \][/tex]
[tex]\[ x=-4 \][/tex]
