Answer :
Sure! Let's break down how to predict the value of [tex]\( y \)[/tex] for a given value of [tex]\( x \)[/tex] using two different methods: a graph of a linear relationship and a table of linear data.
### Using a Graph of a Linear Relationship
1. Plotting the Data Points:
- Start by plotting the data points [tex]\((x, y)\)[/tex] on a coordinate plane (graph paper). Each point represents a pair of values from your dataset.
2. Drawing the Best Fit Line:
- Draw the best fit line through the data points. This line should ideally minimize the distance between it and all the data points. This line represents the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
3. Locating the Given Value of [tex]\( x \)[/tex]:
- Find the given value of [tex]\( x \)[/tex] on the x-axis. Mark this point.
4. Moving Vertically to the Best Fit Line:
- From the marked point on the x-axis, move vertically upwards (or downwards, depending on your graph) until you reach the best fit line.
5. Finding the Corresponding [tex]\( y \)[/tex] Value:
- From the point where you intersect the best fit line, move horizontally to the y-axis. The point where you meet the y-axis is the predicted value of [tex]\( y \)[/tex] for the given [tex]\( x \)[/tex].
6. Reading the Predicted Value:
- Read off the y-axis the value that corresponds to the projection of your given [tex]\( x \)[/tex] value on the best fit line. This is your predicted [tex]\( y \)[/tex] value.
### Using a Table of Linear Data
1. Identifying Closest Data Points:
- Look at your table of linear data and identify the two data points that are closest to the given value of [tex]\( x \)[/tex]. Let's call these points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex].
2. Determining the Slope:
- Calculate the slope [tex]\( m \)[/tex] of the line that connects these two points. The slope is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
3. Writing the Equation of the Line:
- Use the slope and one of the points (it doesn't matter which one) to write the equation of the line. The equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Alternatively, you can use the point-slope form and convert it to the above form:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Solving for [tex]\( y \)[/tex] gives:
[tex]\[ y = m(x - x_1) + y_1 \][/tex]
4. Substituting the Given [tex]\( x \)[/tex] Value:
- Substitute the given [tex]\( x \)[/tex] value into the equation you just derived to solve for the predicted [tex]\( y \)[/tex] value:
[tex]\[ y = m(x - x_1) + y_1 \][/tex]
### Conclusion:
- For the graph method, the predicted value of [tex]\( y \)[/tex] depends on the specific best fit line drawn through the data points and the observation on the graph.
- For the table method, the predicted value of [tex]\( y \)[/tex] depends on the linear interpolation between the closest data points and the specific data present in the table.
In this particular scenario, since we can't visually plot or have specific data points provided, the predicted values for both methods can be represented as placeholders or [tex]\( \text{None} \)[/tex].
### Using a Graph of a Linear Relationship
1. Plotting the Data Points:
- Start by plotting the data points [tex]\((x, y)\)[/tex] on a coordinate plane (graph paper). Each point represents a pair of values from your dataset.
2. Drawing the Best Fit Line:
- Draw the best fit line through the data points. This line should ideally minimize the distance between it and all the data points. This line represents the linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
3. Locating the Given Value of [tex]\( x \)[/tex]:
- Find the given value of [tex]\( x \)[/tex] on the x-axis. Mark this point.
4. Moving Vertically to the Best Fit Line:
- From the marked point on the x-axis, move vertically upwards (or downwards, depending on your graph) until you reach the best fit line.
5. Finding the Corresponding [tex]\( y \)[/tex] Value:
- From the point where you intersect the best fit line, move horizontally to the y-axis. The point where you meet the y-axis is the predicted value of [tex]\( y \)[/tex] for the given [tex]\( x \)[/tex].
6. Reading the Predicted Value:
- Read off the y-axis the value that corresponds to the projection of your given [tex]\( x \)[/tex] value on the best fit line. This is your predicted [tex]\( y \)[/tex] value.
### Using a Table of Linear Data
1. Identifying Closest Data Points:
- Look at your table of linear data and identify the two data points that are closest to the given value of [tex]\( x \)[/tex]. Let's call these points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex].
2. Determining the Slope:
- Calculate the slope [tex]\( m \)[/tex] of the line that connects these two points. The slope is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
3. Writing the Equation of the Line:
- Use the slope and one of the points (it doesn't matter which one) to write the equation of the line. The equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Alternatively, you can use the point-slope form and convert it to the above form:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Solving for [tex]\( y \)[/tex] gives:
[tex]\[ y = m(x - x_1) + y_1 \][/tex]
4. Substituting the Given [tex]\( x \)[/tex] Value:
- Substitute the given [tex]\( x \)[/tex] value into the equation you just derived to solve for the predicted [tex]\( y \)[/tex] value:
[tex]\[ y = m(x - x_1) + y_1 \][/tex]
### Conclusion:
- For the graph method, the predicted value of [tex]\( y \)[/tex] depends on the specific best fit line drawn through the data points and the observation on the graph.
- For the table method, the predicted value of [tex]\( y \)[/tex] depends on the linear interpolation between the closest data points and the specific data present in the table.
In this particular scenario, since we can't visually plot or have specific data points provided, the predicted values for both methods can be represented as placeholders or [tex]\( \text{None} \)[/tex].
