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Sure! Let's determine the gradient and the [tex]\( y \)[/tex]-intercept of the given straight lines.
### 3.1.1. [tex]\( 8x + y = 1 \)[/tex]
First, we'll rewrite the equation in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the gradient and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
Starting from:
[tex]\[ 8x + y = 1 \][/tex]
Subtract [tex]\( 8x \)[/tex] from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = -8x + 1 \][/tex]
From this equation, we can see that:
- The gradient [tex]\( m \)[/tex] is [tex]\(-8\)[/tex]
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( 1 \)[/tex]
So, for [tex]\( 8x + y = 1 \)[/tex]:
- Gradient: [tex]\(-8\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 1 \)[/tex]
### 3.1.2. [tex]\( 1 + 5x - 2y = 0 \)[/tex]
Rewriting the equation in the form [tex]\( y = mx + b \)[/tex]:
Starting from:
[tex]\[ 1 + 5x - 2y = 0 \][/tex]
Rearrange terms to isolate [tex]\( y \)[/tex]:
[tex]\[ -2y = -5x - 1 \][/tex]
Divide every term by [tex]\(-2\)[/tex]:
[tex]\[ y = \frac{5}{2}x + \frac{1}{2} \][/tex]
From this equation, we get:
- The gradient [tex]\( m \)[/tex] is [tex]\( \frac{5}{2} \)[/tex]
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( \frac{1}{2} \)[/tex]
So, for [tex]\( 1 + 5x - 2y = 0 \)[/tex]:
- Gradient: [tex]\( 2.5 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 0.5 \)[/tex]
### 3.1.3. [tex]\( y = \frac{1}{2}x \)[/tex]
This equation is already in the slope-intercept form [tex]\( y = mx + b \)[/tex], where:
- The gradient [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex]
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( 0 \)[/tex] (since there is no constant term)
So, for [tex]\( y = \frac{1}{2}x \)[/tex]:
- Gradient: [tex]\( 0.5 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 0 \)[/tex]
### 3.1.4. [tex]\( 2y + 4x - 6 = 0 \)[/tex]
Rewriting the equation in the form [tex]\( y = mx + b \)[/tex]:
Starting from:
[tex]\[ 2y + 4x - 6 = 0 \][/tex]
Rearrange terms to isolate [tex]\( y \)[/tex]:
[tex]\[ 2y = -4x + 6 \][/tex]
Divide every term by [tex]\( 2 \)[/tex]:
[tex]\[ y = -2x + 3 \][/tex]
From this equation, we understand that:
- The gradient [tex]\( m \)[/tex] is [tex]\(-2\)[/tex]
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( 3 \)[/tex]
So, for [tex]\( 2y + 4x - 6 = 0 \)[/tex]:
- Gradient: [tex]\(-2\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 3 \)[/tex]
In summary:
1. [tex]\( 8x + y = 1 \)[/tex]:
- Gradient: [tex]\(-8\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 1 \)[/tex]
2. [tex]\( 1 + 5x - 2y = 0 \)[/tex]:
- Gradient: [tex]\( 2.5 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 0.5 \)[/tex]
3. [tex]\( y = \frac{1}{2}x \)[/tex]:
- Gradient: [tex]\( 0.5 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 0 \)[/tex]
4. [tex]\( 2y + 4x - 6 = 0 \)[/tex]:
- Gradient: [tex]\(-2\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 3 \)[/tex]
### 3.1.1. [tex]\( 8x + y = 1 \)[/tex]
First, we'll rewrite the equation in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the gradient and [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
Starting from:
[tex]\[ 8x + y = 1 \][/tex]
Subtract [tex]\( 8x \)[/tex] from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = -8x + 1 \][/tex]
From this equation, we can see that:
- The gradient [tex]\( m \)[/tex] is [tex]\(-8\)[/tex]
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( 1 \)[/tex]
So, for [tex]\( 8x + y = 1 \)[/tex]:
- Gradient: [tex]\(-8\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 1 \)[/tex]
### 3.1.2. [tex]\( 1 + 5x - 2y = 0 \)[/tex]
Rewriting the equation in the form [tex]\( y = mx + b \)[/tex]:
Starting from:
[tex]\[ 1 + 5x - 2y = 0 \][/tex]
Rearrange terms to isolate [tex]\( y \)[/tex]:
[tex]\[ -2y = -5x - 1 \][/tex]
Divide every term by [tex]\(-2\)[/tex]:
[tex]\[ y = \frac{5}{2}x + \frac{1}{2} \][/tex]
From this equation, we get:
- The gradient [tex]\( m \)[/tex] is [tex]\( \frac{5}{2} \)[/tex]
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( \frac{1}{2} \)[/tex]
So, for [tex]\( 1 + 5x - 2y = 0 \)[/tex]:
- Gradient: [tex]\( 2.5 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 0.5 \)[/tex]
### 3.1.3. [tex]\( y = \frac{1}{2}x \)[/tex]
This equation is already in the slope-intercept form [tex]\( y = mx + b \)[/tex], where:
- The gradient [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex]
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( 0 \)[/tex] (since there is no constant term)
So, for [tex]\( y = \frac{1}{2}x \)[/tex]:
- Gradient: [tex]\( 0.5 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 0 \)[/tex]
### 3.1.4. [tex]\( 2y + 4x - 6 = 0 \)[/tex]
Rewriting the equation in the form [tex]\( y = mx + b \)[/tex]:
Starting from:
[tex]\[ 2y + 4x - 6 = 0 \][/tex]
Rearrange terms to isolate [tex]\( y \)[/tex]:
[tex]\[ 2y = -4x + 6 \][/tex]
Divide every term by [tex]\( 2 \)[/tex]:
[tex]\[ y = -2x + 3 \][/tex]
From this equation, we understand that:
- The gradient [tex]\( m \)[/tex] is [tex]\(-2\)[/tex]
- The [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is [tex]\( 3 \)[/tex]
So, for [tex]\( 2y + 4x - 6 = 0 \)[/tex]:
- Gradient: [tex]\(-2\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 3 \)[/tex]
In summary:
1. [tex]\( 8x + y = 1 \)[/tex]:
- Gradient: [tex]\(-8\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 1 \)[/tex]
2. [tex]\( 1 + 5x - 2y = 0 \)[/tex]:
- Gradient: [tex]\( 2.5 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 0.5 \)[/tex]
3. [tex]\( y = \frac{1}{2}x \)[/tex]:
- Gradient: [tex]\( 0.5 \)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 0 \)[/tex]
4. [tex]\( 2y + 4x - 6 = 0 \)[/tex]:
- Gradient: [tex]\(-2\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\( 3 \)[/tex]
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