Question

Does the equation represent a linear or non linear function?

1. y=x+5
2. y=\frac{4x}{3}
3. y=1-x^2

Answer 1

Let's examine each equation to determine whether it represents a linear or non-linear function.

1. \( y = x + 5 \)
- This equation is in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Since there are no exponents on \( x \) other than 1, and \( x \) is not involved in any other operations like multiplication by itself, division, or non-linear functions, this is a linear equation.

2. \( y = \frac{4x}{3} \)
- This can be rewritten as \( y = \left(\frac{4}{3}\right)x \), which again is in the form \( y = mx + b \) with \( m = \frac{4}{3} \) and \( b = 0 \). This is clearly a linear equation.

3. \( y = 1 - x^2 \)
- This equation includes an \( x^2 \) term, which means the power of \( x \) is 2. The presence of this squared term makes the function non-linear. The graph of this equation would be a parabola, indicating a quadratic relationship.

So, summarizing:

1. \( y = x + 5 \) is linear.
2. \( y = \frac{4x}{3} \) is linear.
3. \( y = 1 - x^2 \) is non-linear.