Sure! Let's determine the gradients (slopes) and [tex]\( y \)[/tex]-intercepts for each equation step-by-step.
1. Equation [tex]\( y = 5x + 5 \)[/tex]:
- The equation is already in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the gradient and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
- Here, the gradient [tex]\( m \)[/tex] is [tex]\( 5 \)[/tex].
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( 5 \)[/tex].
So, for [tex]\( y = 5x + 5 \)[/tex]:
- Gradient: [tex]\( 5 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 5 \)[/tex]
2. Equation [tex]\( y = \frac{4}{2}x + 4 \)[/tex]:
- Simplify the fraction [tex]\( \frac{4}{2} \)[/tex]:
[tex]\[
\frac{4}{2} = 2
\][/tex]
- So, the equation becomes [tex]\( y = 2x + 4 \)[/tex].
- In this form, the gradient [tex]\( m \)[/tex] is [tex]\( 2 \)[/tex].
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( 4 \)[/tex].
So, for [tex]\( y = 2x + 4 \)[/tex]:
- Gradient: [tex]\( 2 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 4 \)[/tex]
3. Equation [tex]\( y = -\frac{1}{3}x - 1 \)[/tex]:
- This equation is already in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the gradient and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
- Here, the gradient [tex]\( m \)[/tex] is [tex]\( -\frac{1}{3} \)[/tex].
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( -1 \)[/tex].
So, for [tex]\( y = -\frac{1}{3}x - 1 \)[/tex]:
- Gradient: [tex]\( -\frac{1}{3} \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( -1 \)[/tex]
Now, let's fill in the table with the gradients and [tex]\( y \)[/tex]-intercepts:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Equation} & \text{Gradient} & y\text{-intercept} \\
\hline
y = 5x + 5 & 5 & 5 \\
\hline
y = \frac{4}{2}x + 4 & 2 & 4 \\
\hline
y = -\frac{1}{3}x - 1 & -\frac{1}{3} & -1 \\
\hline
\end{array}
\][/tex]
