To find the equations of the lines given their gradients (slopes) and [tex]\(y\)[/tex]-intercepts, we use the slope-intercept form of a line equation, which is:
[tex]\[ y = mx + c \][/tex]
Here:
- [tex]\( m \)[/tex] is the gradient (slope) of the line.
- [tex]\( c \)[/tex] is the [tex]\( y \)[/tex]-intercept (the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]).
Let's determine each equation step-by-step:
1. First Line:
- Gradient ([tex]\( m \)[/tex]) = 3
- [tex]\( y \)[/tex]-intercept ([tex]\( c \)[/tex]) = 2
Substituting these values into the slope-intercept form:
[tex]\[ y = 3x + 2 \][/tex]
2. Second Line:
- Gradient ([tex]\( m \)[/tex]) = -5
- [tex]\( y \)[/tex]-intercept ([tex]\( c \)[/tex]) = 1
Substituting these values into the slope-intercept form:
[tex]\[ y = -5x + 1 \][/tex]
3. Third Line:
- Gradient ([tex]\( m \)[/tex]) = \frac{16}{4} = 4.0
- [tex]\( y \)[/tex]-intercept ([tex]\( c \)[/tex]) = -6
Substituting these values into the slope-intercept form:
[tex]\[ y = 4.0x - 6 \][/tex]
So, filling in the equations in the table, we get:
\begin{tabular}{|c|c|c|}
\hline Gradient & [tex]$y$[/tex]-intercept & Equation \\
\hline 3 & [tex]$(0,2)$[/tex] & [tex]\( y = 3x + 2 \)[/tex] \\
\hline-5 & [tex]$(0,1)$[/tex] & [tex]\( y = -5x + 1 \)[/tex] \\
\hline 4.0 & [tex]$(0,-6)$[/tex] & [tex]\( y = 4.0x - 6 \)[/tex] \\
\hline
\end{tabular}
These are the equations of the lines based on the given gradients and [tex]\( y \)[/tex]-intercepts.
