Answered

Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

Select all of the equations that represent linear relationships.

A. [tex]5 + 2y = 13[/tex]
B. [tex]y = \frac{1}{2} x^2 + 7[/tex]
C. [tex]y - 5 = 2(x - 1)[/tex]
D. [tex]\frac{y}{2} = x + 7[/tex]
E. [tex]x = -4[/tex]

Sagot :

GinnyAnswer
Let's determine which of the given equations represent linear relationships.

### Equation 1: [tex]\( 5 + 2y = 13 \)[/tex]

First, we rearrange it to solve for [tex]\( y \)[/tex]:
[tex]\[ 5 + 2y = 13 \][/tex]
Subtract 5 from both sides:
[tex]\[ 2y = 8 \][/tex]
Divide both sides by 2:
[tex]\[ y = 4 \][/tex]

This equation is in the form [tex]\( y = mx + b \)[/tex] where [tex]\( m = 0 \)[/tex] and [tex]\( b = 4 \)[/tex], which is a linear equation.

### Equation 2: [tex]\( y = \frac{1}{2}x^2 + 7 \)[/tex]

This equation includes [tex]\( x^2 \)[/tex], which makes it a quadratic equation. Since it's not in the form [tex]\( y = mx + b \)[/tex], it is not a linear equation.

### Equation 3: [tex]\( y - 5 = 2(x - 1) \)[/tex]

First, let's simplify and rearrange this equation:
[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 equation is in the form [tex]\( y = mx + b \)[/tex] where [tex]\( m = 2 \)[/tex] and [tex]\( b = 3 \)[/tex], which is a linear equation.

### Equation 4: [tex]\( \frac{y}{2} = x + 7 \)[/tex]

First, let's solve for [tex]\( y \)[/tex]:
[tex]\[ \frac{y}{2} = x + 7 \][/tex]
Multiply both sides by 2:
[tex]\[ y = 2(x + 7) \][/tex]
[tex]\[ y = 2x + 14 \][/tex]

This equation is in the form [tex]\( y = mx + b \)[/tex] where [tex]\( m = 2 \)[/tex] and [tex]\( b = 14 \)[/tex], which is a linear equation.

### Equation 5: [tex]\( x = -4 \)[/tex]

This equation does not have a [tex]\( y \)[/tex] variable and thus represents a vertical line. Even though it is a linear equation in one variable, for the purposes of identifying linear relationships typically involving both [tex]\( x \)[/tex] and [tex]\( y \)[/tex], it is considered linear in the sense that it maintains the constant.

### Summary

After inspecting all the equations, the ones that are linear are:
[tex]\[ 5 + 2y = 13 \][/tex]
[tex]\[ y - 5 = 2(x - 1) \][/tex]
[tex]\[ \frac{y}{2} = x + 7 \][/tex]
[tex]\[ x = -4 \][/tex]

Thus, the equations representing linear relationships are:
[tex]\[ 5 + 2y = 13 \][/tex]
[tex]\[ y - 5 = 2(x - 1) \][/tex]
[tex]\[ \frac{y}{2} = x + 7 \][/tex]
[tex]\[ x = -4 \][/tex]

The correct indices are:
[tex]\[ [0, 2, 3, 4] \][/tex]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.