Question

for each equation, determine whether it is linear.

equation	is the equation linear?
(b) $y = x^2 - 6$	$\circ$	$\circ$
(c) $y = -2x$	$\circ$	$\circ$
(d) $y = -x + 2$	$\circ$	$\circ$

Explanation:

To determine if an equation is linear, we use the definition of a linear equation in two variables, which is of the form \( y = mx + b \), where \( m \) and \( b \) are constants, and the highest power of \( x \) is 1.

Part (a)

• Step 1: Analyze the equation \( y = x^3 \).

The highest power of \( x \) is 3, which is not 1.

• Conclusion: Not linear.

Part (b)

• Step 1: Analyze the equation \( y = x^2 - 6 \).

The highest power of \( x \) is 2, which is not 1.

• Conclusion: Not linear.

Part (c)

• Step 1: Analyze the equation \( y = -2x \).

This can be written as \( y = -2x + 0 \), which matches the linear form \( y = mx + b \) (with \( m = -2 \) and \( b = 0 \)). The highest power of \( x \) is 1.

• Conclusion: Linear.

Part (d)

• Step 1: Analyze the equation \( y = -x + 2 \).

This matches the linear form \( y = mx + b \) (with \( m = -1 \) and \( b = 2 \)). The highest power of \( x \) is 1.

• Conclusion: Linear.

Final Answers:

Equation	Is the equation linear?
(b) \( y = x^2 - 6 \)	No
(c) \( y = -2x \)	Yes
(d) \( y = -x + 2 \)	Yes

For the table:

• (a): Select "No"
• (b): Select "No"
• (c): Select "Yes"
• (d): Select "Yes"

Answer:

To determine if an equation is linear, we use the definition of a linear equation in two variables, which is of the form \( y = mx + b \), where \( m \) and \( b \) are constants, and the highest power of \( x \) is 1.

Part (a)

• Step 1: Analyze the equation \( y = x^3 \).

The highest power of \( x \) is 3, which is not 1.

• Conclusion: Not linear.

Part (b)

• Step 1: Analyze the equation \( y = x^2 - 6 \).

The highest power of \( x \) is 2, which is not 1.

• Conclusion: Not linear.

Part (c)

• Step 1: Analyze the equation \( y = -2x \).

This can be written as \( y = -2x + 0 \), which matches the linear form \( y = mx + b \) (with \( m = -2 \) and \( b = 0 \)). The highest power of \( x \) is 1.

• Conclusion: Linear.

Part (d)

• Step 1: Analyze the equation \( y = -x + 2 \).

This matches the linear form \( y = mx + b \) (with \( m = -1 \) and \( b = 2 \)). The highest power of \( x \) is 1.

• Conclusion: Linear.

Final Answers:

Equation	Is the equation linear?
(b) \( y = x^2 - 6 \)	No
(c) \( y = -2x \)	Yes
(d) \( y = -x + 2 \)	Yes

For the table:

• (a): Select "No"
• (b): Select "No"
• (c): Select "Yes"
• (d): Select "Yes"