Let's start by rewriting the given linear equation in slope-intercept form, which is [tex]\( y = mx + b \)[/tex].
The given equation is:
[tex]\[ 7x + 4y = 17 \][/tex]
To convert this into slope-intercept form, we need to solve for [tex]\( y \)[/tex].
1. First, isolate the [tex]\( y \)[/tex]-term:
[tex]\[ 4y = 17 - 7x \][/tex]
2. Next, solve for [tex]\( y \)[/tex] by dividing every term by 4:
[tex]\[ y = \frac{17}{4} - \frac{7}{4}x \][/tex]
Rewriting it in standard slope-intercept form [tex]\( y = mx + b \)[/tex], we have:
[tex]\[ y = -\frac{7}{4}x + \frac{17}{4} \][/tex]
In this form, [tex]\( m \)[/tex] represents the slope and [tex]\( b \)[/tex] represents the y-intercept.
- So, the slope [tex]\( m \)[/tex] is [tex]\( -\frac{7}{4} \)[/tex].
Next, identify the intercepts:
1. Vertical [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex]:
The [tex]\( y \)[/tex]-intercept is the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 0. From our slope-intercept form:
[tex]\[ y = -\frac{7}{4}(0) + \frac{17}{4} \][/tex]
[tex]\[ y = \frac{17}{4} \][/tex]
Therefore, the vertical [tex]\( y \)[/tex]-intercept is [tex]\( (0, \frac{17}{4}) \)[/tex].
2. Horizontal [tex]\( x \)[/tex]-intercept:
The [tex]\( x \)[/tex]-intercept is the value of [tex]\( x \)[/tex] when [tex]\( y \)[/tex] is 0. We use the original equation [tex]\( 7x + 4y = 17 \)[/tex]:
[tex]\[ 7x + 4(0) = 17 \][/tex]
[tex]\[ 7x = 17 \][/tex]
[tex]\[ x = \frac{17}{7} \][/tex]
Hence, the horizontal [tex]\( x \)[/tex]-intercept is [tex]\( (\frac{17}{7}, 0) \)[/tex].
Summarizing:
- The slope [tex]\( m \)[/tex] is [tex]\( -\frac{7}{4} \)[/tex].
- The vertical [tex]\( y \)[/tex]-intercept is [tex]\( (0, \frac{17}{4}) \)[/tex].
- The horizontal [tex]\( x \)[/tex]-intercept is [tex]\( (\frac{17}{7}, 0) \)[/tex].
According to the corrected answer:
- Slope [tex]\( m = -\frac{7}{4} \)[/tex]
- Vertical [tex]\( y \)[/tex]-intercept: [tex]\( (0, \frac{17}{4}) \)[/tex]
- Horizontal [tex]\( x \)[/tex]-intercept: [tex]\( (\frac{17}{7}, 0) \)[/tex]
The equation [tex]\( y = 4x + 7 \)[/tex] was a misstatement in the problem, as it does not align with the given equation [tex]\( 7x + 4y = 17 \)[/tex].
