To work out the gradient of each line, let's follow a step-by-step process for each equation.
### Finding the Gradient of the Line [tex]\( y = 5x - 3 \)[/tex]
1. Identify the form of the equation:
- The given equation is [tex]\( y = 5x - 3 \)[/tex].
- This is in the slope-intercept form, [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the gradient.
2. Determine the gradient:
- By comparing [tex]\( y = 5x - 3 \)[/tex] to [tex]\( y = mx + c \)[/tex], we see that [tex]\( m = 5 \)[/tex].
Therefore, the gradient of the line [tex]\( y = 5x - 3 \)[/tex] is [tex]\( 5 \)[/tex].
### Finding the Gradient of the Line [tex]\( 3y - 12x + 7 = 0 \)[/tex]
1. Rearrange the equation into the slope-intercept form:
- Start with [tex]\( 3y - 12x + 7 = 0 \)[/tex].
- Add [tex]\( 12x \)[/tex] to both sides: [tex]\( 3y = 12x - 7 \)[/tex].
- Divide by 3: [tex]\( y = 4x - \frac{7}{3} \)[/tex].
2. Identify the gradient:
- The equation is now in the form [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the gradient.
- Here, [tex]\( m = 4 \)[/tex].
Therefore, the gradient of the line [tex]\( 3y - 12x + 7 = 0 \)[/tex] is [tex]\( 4 \)[/tex].
Summarizing the gradients:
- The gradient of the line [tex]\( y = 5x - 3 \)[/tex] is [tex]\( 5 \)[/tex].
- The gradient of the line [tex]\( 3y - 12x + 7 = 0 \)[/tex] is [tex]\( 4 \)[/tex].
