Sure, let's work through finding the equation of the line step-by-step.
### Step 1: Understand the form of the line equation
We want to find the equation of a line that has a gradient (slope) of 3 and passes through the point [tex]\((-1, 3)\)[/tex]. The general form of a linear equation is [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
### Step 2: Use the point-slope form of the line equation
We can also use the point-slope form of the equation of a line, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\( m \)[/tex] is the slope.
For this problem:
- The gradient [tex]\( m = 3 \)[/tex]
- The given point is [tex]\((-1, 3)\)[/tex], so [tex]\( x_1 = -1 \)[/tex] and [tex]\( y_1 = 3 \)[/tex]
### Step 3: Substitute these values into the point-slope form
Substitute the values into the point-slope form equation:
[tex]\[ y - 3 = 3(x - (-1)) \][/tex]
### Step 4: Simplify the equation
First, simplify the expression in the parentheses:
[tex]\[ y - 3 = 3(x + 1) \][/tex]
Now distribute the 3 across the terms in the parentheses:
[tex]\[ y - 3 = 3x + 3 \][/tex]
Next, solve for [tex]\( y \)[/tex] to get the equation in the slope-intercept form [tex]\( y = mx + c \)[/tex]:
[tex]\[ y = 3x + 3 + 3 \][/tex]
[tex]\[ y = 3x + 6 \][/tex]
### Final Result
Hence, the equation of the line with a gradient of 3 that passes through the point [tex]\((-1, 3)\)[/tex] is:
[tex]\[ y = 3x + 6 \][/tex]
