Answer :
To find the equation of a line with a given slope \( m = 3 \) that passes through the point \( (7, 5) \), we can use the point-slope form of a linear equation. The point-slope form is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope of the line. Here, \( (x_1, y_1) = (7, 5) \) and \( m = 3 \).
Step 1: Write the point-slope form using the given values:
[tex]\[ y - 5 = 3(x - 7) \][/tex]
Step 2: Distribute the slope \( m = 3 \) on the right-hand side:
[tex]\[ y - 5 = 3x - 21 \][/tex]
Step 3: Solve for \( y \) to put the equation into the slope-intercept form \( y = mx + b \):
[tex]\[ y = 3x - 21 + 5 \][/tex]
[tex]\[ y = 3x - 16 \][/tex]
Therefore, the equation of the line in the form \( y = mx + b \) is:
[tex]\[ y = 3x - 16 \][/tex]
The \( y \)-intercept \( b \) is \(-16\).
So, the final equation of the line is:
[tex]\[ y = 3x - 16 \][/tex]
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope of the line. Here, \( (x_1, y_1) = (7, 5) \) and \( m = 3 \).
Step 1: Write the point-slope form using the given values:
[tex]\[ y - 5 = 3(x - 7) \][/tex]
Step 2: Distribute the slope \( m = 3 \) on the right-hand side:
[tex]\[ y - 5 = 3x - 21 \][/tex]
Step 3: Solve for \( y \) to put the equation into the slope-intercept form \( y = mx + b \):
[tex]\[ y = 3x - 21 + 5 \][/tex]
[tex]\[ y = 3x - 16 \][/tex]
Therefore, the equation of the line in the form \( y = mx + b \) is:
[tex]\[ y = 3x - 16 \][/tex]
The \( y \)-intercept \( b \) is \(-16\).
So, the final equation of the line is:
[tex]\[ y = 3x - 16 \][/tex]
