To solve the given problem, we need to convert the equation [tex]\( 2x + 3y = 1470 \)[/tex] into the slope-intercept form, which is written as [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope and [tex]\( b \)[/tex] represents the y-intercept.
Here's a detailed, step-by-step process to convert the given equation:
### Step 1: Start with the given equation:
[tex]\[ 2x + 3y = 1470 \][/tex]
### Step 2: Isolate the term involving [tex]\( y \)[/tex]:
To do this, subtract [tex]\( 2x \)[/tex] from both sides of the equation:
[tex]\[ 3y = -2x + 1470 \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]:
Divide every term by 3 to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-2x}{3} + \frac{1470}{3} \][/tex]
### Step 4: Simplify the fractions:
[tex]\[ y = -\frac{2}{3}x + 490 \][/tex]
Now the equation is in slope-intercept form [tex]\( y = mx + b \)[/tex].
### Identifying the slope and y-intercept:
- The slope [tex]\( m \)[/tex] is the coefficient of [tex]\( x \)[/tex], which is [tex]\( -\frac{2}{3} \)[/tex].
- The y-intercept [tex]\( b \)[/tex] is the constant term, which is [tex]\( 490 \)[/tex].
So, the slope of the line is [tex]\( -0.6666666666666666 \)[/tex] (or [tex]\( -\frac{2}{3} \)[/tex] in fractional form) and the y-intercept is [tex]\( 490 \)[/tex].
Thus, the slope-intercept form of the equation [tex]\( 2x + 3y = 1470 \)[/tex] is:
[tex]\[ y = -\frac{2}{3}x + 490 \][/tex]
The slope [tex]\( m \)[/tex] is [tex]\( -0.6666666666666666 \)[/tex] and the y-intercept [tex]\( b \)[/tex] is [tex]\( 490 \)[/tex].
