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Sagot :
Certainly! Let's translate the given statement into a mathematical equation, and then graph this equation step-by-step.
1. Understanding the Statement:
- The statement given is: "The y-value is 6 less than the x-value."
- This indicates a relationship between the variables [tex]\( y \)[/tex] and [tex]\( x \)[/tex].
2. Writing the Equation:
- To translate this into a mathematical equation, we simply follow the statement.
- If [tex]\( y \)[/tex] is 6 less than [tex]\( x \)[/tex], then we can write:
[tex]\[ y = x - 6 \][/tex]
So the equation in two variables is:
[tex]\[ y = x - 6 \][/tex]
3. Graphing the Equation:
- To graph the equation [tex]\( y = x - 6 \)[/tex], we need to plot points that satisfy this equation and then connect those points.
- Let's pick a few values for [tex]\( x \)[/tex] and compute the corresponding [tex]\( y \)[/tex] values.
[tex]\[ \begin{array}{c|c} x & y \\ \hline -2 & -2 - 6 = -8 \\ 0 & 0 - 6 = -6 \\ 2 & 2 - 6 = -4 \\ 4 & 4 - 6 = -2 \\ 6 & 6 - 6 = 0 \\ 8 & 8 - 6 = 2 \\ 10 & 10 - 6 = 4 \\ \end{array} \][/tex]
4. Plotting the Points:
- Let's plot these points on a coordinate system:
[tex]\((-2, -8)\)[/tex], [tex]\((0, -6)\)[/tex], [tex]\((2, -4)\)[/tex], [tex]\((4, -2)\)[/tex], [tex]\((6, 0)\)[/tex], [tex]\((8, 2)\)[/tex], [tex]\((10, 4)\)[/tex]
5. Drawing the Line:
- Once the points are plotted, draw a straight line through these points. Since the equation is linear (in the form [tex]\( y = mx + b \)[/tex]), the graph will be a straight line.
6. Graph:
Here's how you can graph the equation manually or visualize it on a graph paper or using graphing tools/software:
- The y-intercept is at [tex]\( (0, -6) \)[/tex], where the line crosses the y-axis.
- The slope (rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]) is 1, meaning for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 1 unit.
### Graph of the equation [tex]\( y = x - 6 \)[/tex]:
[tex]\[ \begin{array}{c|c} x & y \\ \hline 0 & -6 \\ 1 & -5 \\ 2 & -4 \\ \vdots & \vdots \\ 10 & 4 \\ \end{array} \][/tex]
This creates a straight line with a positive slope passing through the point (0, -6).
On a coordinate plane, the line extends infinitely in both directions but passes through all the points calculated based on the equation [tex]\( y = x - 6 \)[/tex].
In summary, the equation derived from the given statement is [tex]\( y = x - 6 \)[/tex], and its graph is a straight line intersecting the y-axis at [tex]\((0, -6)\)[/tex] and having a slope of 1.
1. Understanding the Statement:
- The statement given is: "The y-value is 6 less than the x-value."
- This indicates a relationship between the variables [tex]\( y \)[/tex] and [tex]\( x \)[/tex].
2. Writing the Equation:
- To translate this into a mathematical equation, we simply follow the statement.
- If [tex]\( y \)[/tex] is 6 less than [tex]\( x \)[/tex], then we can write:
[tex]\[ y = x - 6 \][/tex]
So the equation in two variables is:
[tex]\[ y = x - 6 \][/tex]
3. Graphing the Equation:
- To graph the equation [tex]\( y = x - 6 \)[/tex], we need to plot points that satisfy this equation and then connect those points.
- Let's pick a few values for [tex]\( x \)[/tex] and compute the corresponding [tex]\( y \)[/tex] values.
[tex]\[ \begin{array}{c|c} x & y \\ \hline -2 & -2 - 6 = -8 \\ 0 & 0 - 6 = -6 \\ 2 & 2 - 6 = -4 \\ 4 & 4 - 6 = -2 \\ 6 & 6 - 6 = 0 \\ 8 & 8 - 6 = 2 \\ 10 & 10 - 6 = 4 \\ \end{array} \][/tex]
4. Plotting the Points:
- Let's plot these points on a coordinate system:
[tex]\((-2, -8)\)[/tex], [tex]\((0, -6)\)[/tex], [tex]\((2, -4)\)[/tex], [tex]\((4, -2)\)[/tex], [tex]\((6, 0)\)[/tex], [tex]\((8, 2)\)[/tex], [tex]\((10, 4)\)[/tex]
5. Drawing the Line:
- Once the points are plotted, draw a straight line through these points. Since the equation is linear (in the form [tex]\( y = mx + b \)[/tex]), the graph will be a straight line.
6. Graph:
Here's how you can graph the equation manually or visualize it on a graph paper or using graphing tools/software:
- The y-intercept is at [tex]\( (0, -6) \)[/tex], where the line crosses the y-axis.
- The slope (rate of change of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]) is 1, meaning for every 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 1 unit.
### Graph of the equation [tex]\( y = x - 6 \)[/tex]:
[tex]\[ \begin{array}{c|c} x & y \\ \hline 0 & -6 \\ 1 & -5 \\ 2 & -4 \\ \vdots & \vdots \\ 10 & 4 \\ \end{array} \][/tex]
This creates a straight line with a positive slope passing through the point (0, -6).
On a coordinate plane, the line extends infinitely in both directions but passes through all the points calculated based on the equation [tex]\( y = x - 6 \)[/tex].
In summary, the equation derived from the given statement is [tex]\( y = x - 6 \)[/tex], and its graph is a straight line intersecting the y-axis at [tex]\((0, -6)\)[/tex] and having a slope of 1.
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