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Sagot :
Let's analyze each of the given relations step-by-step to determine their gradients and y-intercepts.
### Part a: [tex]\( y = 2x - 4 \)[/tex]
This equation is already in the slope-intercept form, which is [tex]\( y = mx + b \)[/tex].
- The slope (or gradient) is the coefficient of [tex]\( x \)[/tex]. Here, [tex]\( m = 2 \)[/tex].
- The y-intercept is the constant term, which is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]. Here, [tex]\( b = -4 \)[/tex].
So for [tex]\( y = 2x - 4 \)[/tex]:
- Gradient (slope) = 2
- Y-intercept = -4
### Part b: [tex]\( x + 3y = 6 \)[/tex]
For this equation, we need to rearrange it into the slope-intercept form [tex]\( y = mx + b \)[/tex].
Starting from:
[tex]\[ x + 3y = 6 \][/tex]
Step 1: Isolate the [tex]\( y \)[/tex]-term by moving [tex]\( x \)[/tex] to the other side:
[tex]\[ 3y = -x + 6 \][/tex]
Step 2: Solve for [tex]\( y \)[/tex] by dividing every term by 3:
[tex]\[ y = -\frac{1}{3}x + 2 \][/tex]
Now, the equation is in slope-intercept form [tex]\( y = mx + b \)[/tex]:
- The slope (or gradient) is the coefficient of [tex]\( x \)[/tex]. Here, [tex]\( m = -\frac{1}{3} \)[/tex].
- The y-intercept is the constant term, which is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]. Here, [tex]\( b = 2 \)[/tex].
So for [tex]\( x + 3y = 6 \)[/tex]:
- Gradient (slope) = -[tex]\(\frac{1}{3}\)[/tex]
- Y-intercept = 2
### Summary of Results
1. For [tex]\( y = 2x - 4 \)[/tex]:
- Gradient (slope) = 2
- Y-intercept = -4
2. For [tex]\( x + 3y = 6 \)[/tex]:
- Gradient (slope) = -[tex]\(\frac{1}{3}\)[/tex]
- Y-intercept = 2
### Sketching the Graphs
To sketch the graphs of these two equations:
1. [tex]\( y = 2x - 4 \)[/tex]:
- Start at the y-intercept (-4) on the y-axis.
- From this point, use the slope to determine the next point. Since the slope is 2, for every 1 unit you move to the right on the x-axis, move 2 units up.
2. [tex]\( x + 3y = 6 \)[/tex]:
- Start at the y-intercept (2) on the y-axis.
- From this point, use the slope to determine the next point. Since the slope is -[tex]\(\frac{1}{3}\)[/tex], for every 3 units you move to the right on the x-axis, move 1 unit down. Alternatively, for every 1 unit you move to the right, move [tex]\( \frac{1}{3} \)[/tex] unit down.
By plotting these points and drawing the lines through them, you will obtain the respective graphs of the given relations.
### Part a: [tex]\( y = 2x - 4 \)[/tex]
This equation is already in the slope-intercept form, which is [tex]\( y = mx + b \)[/tex].
- The slope (or gradient) is the coefficient of [tex]\( x \)[/tex]. Here, [tex]\( m = 2 \)[/tex].
- The y-intercept is the constant term, which is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]. Here, [tex]\( b = -4 \)[/tex].
So for [tex]\( y = 2x - 4 \)[/tex]:
- Gradient (slope) = 2
- Y-intercept = -4
### Part b: [tex]\( x + 3y = 6 \)[/tex]
For this equation, we need to rearrange it into the slope-intercept form [tex]\( y = mx + b \)[/tex].
Starting from:
[tex]\[ x + 3y = 6 \][/tex]
Step 1: Isolate the [tex]\( y \)[/tex]-term by moving [tex]\( x \)[/tex] to the other side:
[tex]\[ 3y = -x + 6 \][/tex]
Step 2: Solve for [tex]\( y \)[/tex] by dividing every term by 3:
[tex]\[ y = -\frac{1}{3}x + 2 \][/tex]
Now, the equation is in slope-intercept form [tex]\( y = mx + b \)[/tex]:
- The slope (or gradient) is the coefficient of [tex]\( x \)[/tex]. Here, [tex]\( m = -\frac{1}{3} \)[/tex].
- The y-intercept is the constant term, which is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]. Here, [tex]\( b = 2 \)[/tex].
So for [tex]\( x + 3y = 6 \)[/tex]:
- Gradient (slope) = -[tex]\(\frac{1}{3}\)[/tex]
- Y-intercept = 2
### Summary of Results
1. For [tex]\( y = 2x - 4 \)[/tex]:
- Gradient (slope) = 2
- Y-intercept = -4
2. For [tex]\( x + 3y = 6 \)[/tex]:
- Gradient (slope) = -[tex]\(\frac{1}{3}\)[/tex]
- Y-intercept = 2
### Sketching the Graphs
To sketch the graphs of these two equations:
1. [tex]\( y = 2x - 4 \)[/tex]:
- Start at the y-intercept (-4) on the y-axis.
- From this point, use the slope to determine the next point. Since the slope is 2, for every 1 unit you move to the right on the x-axis, move 2 units up.
2. [tex]\( x + 3y = 6 \)[/tex]:
- Start at the y-intercept (2) on the y-axis.
- From this point, use the slope to determine the next point. Since the slope is -[tex]\(\frac{1}{3}\)[/tex], for every 3 units you move to the right on the x-axis, move 1 unit down. Alternatively, for every 1 unit you move to the right, move [tex]\( \frac{1}{3} \)[/tex] unit down.
By plotting these points and drawing the lines through them, you will obtain the respective graphs of the given relations.
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