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Sagot :
To determine which statement is true, we need to find the [tex]$y$[/tex]-values of Function A and Function B when [tex]$x = 10$[/tex].
### Step 1: Determine the equation of Function A
We know that Function A is linear and we are given three points: [tex]$(-3, -5)$[/tex], [tex]$(-2, -4)$[/tex], and [tex]$(8, 6)$[/tex]. We can use two of these points to find the slope and the y-intercept of the function.
- Finding the slope (m):
Using the points [tex]$(-3, -5)$[/tex] and [tex]$(-2, -4)$[/tex], we use the slope formula:
[tex]\[ \text{slope (m)} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the points into the formula:
[tex]\[ m = \frac{-4 - (-5)}{-2 - (-3)} = \frac{-4 + 5}{-2 + 3} = \frac{1}{1} = 1 \][/tex]
- Finding the y-intercept (b):
Now that we know the slope [tex]$m = 1$[/tex], we can use one of the points and the slope-intercept form [tex]$y = mx + b$[/tex] to find the y-intercept [tex]$b$[/tex]. Let's use the point [tex]$(-3, -5)$[/tex].
Substitute into the equation:
[tex]\[ -5 = 1(-3) + b \][/tex]
[tex]\[ -5 = -3 + b \][/tex]
[tex]\[ b = -5 + 3 \][/tex]
[tex]\[ b = -2 \][/tex]
So, the equation of Function A is:
[tex]\[ y = 1x - 2 \][/tex]
or simply,
[tex]\[ y = x - 2 \][/tex]
### Step 2: Calculate the y-value of Function A when [tex]$x = 10$[/tex]
Using the equation [tex]$y = x - 2$[/tex], substitute [tex]$x = 10$[/tex]:
[tex]\[ y = 10 - 2 \][/tex]
[tex]\[ y = 8 \][/tex]
So, the [tex]$y$[/tex]-value of Function A when [tex]$x = 10$[/tex] is 8.
### Step 3: Calculate the y-value of Function B when [tex]$x = 10$[/tex]
We are given the equation for Function B as:
[tex]\[ y = \frac{1}{2}x - 3 \][/tex]
Substitute [tex]$x = 10$[/tex] into this equation:
[tex]\[ y = \frac{1}{2}(10) - 3 \][/tex]
[tex]\[ y = 5 - 3 \][/tex]
[tex]\[ y = 2 \][/tex]
So, the [tex]$y$[/tex]-value of Function B when [tex]$x = 10$[/tex] is 2.
### Step 4: Compare the y-values of Function A and Function B when [tex]$x = 10$[/tex]
- The [tex]$y$[/tex]-value of Function A when [tex]$x = 10$[/tex] is 8.
- The [tex]$y$[/tex]-value of Function B when [tex]$x = 10$[/tex] is 2.
Since 8 is greater than 2, the statement that is true is:
The [tex]$y$[/tex]-value of Function A when [tex]$x = 10$[/tex] is greater than the [tex]$y$[/tex]-value of Function B when [tex]$x = 10$[/tex].
### Step 1: Determine the equation of Function A
We know that Function A is linear and we are given three points: [tex]$(-3, -5)$[/tex], [tex]$(-2, -4)$[/tex], and [tex]$(8, 6)$[/tex]. We can use two of these points to find the slope and the y-intercept of the function.
- Finding the slope (m):
Using the points [tex]$(-3, -5)$[/tex] and [tex]$(-2, -4)$[/tex], we use the slope formula:
[tex]\[ \text{slope (m)} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the points into the formula:
[tex]\[ m = \frac{-4 - (-5)}{-2 - (-3)} = \frac{-4 + 5}{-2 + 3} = \frac{1}{1} = 1 \][/tex]
- Finding the y-intercept (b):
Now that we know the slope [tex]$m = 1$[/tex], we can use one of the points and the slope-intercept form [tex]$y = mx + b$[/tex] to find the y-intercept [tex]$b$[/tex]. Let's use the point [tex]$(-3, -5)$[/tex].
Substitute into the equation:
[tex]\[ -5 = 1(-3) + b \][/tex]
[tex]\[ -5 = -3 + b \][/tex]
[tex]\[ b = -5 + 3 \][/tex]
[tex]\[ b = -2 \][/tex]
So, the equation of Function A is:
[tex]\[ y = 1x - 2 \][/tex]
or simply,
[tex]\[ y = x - 2 \][/tex]
### Step 2: Calculate the y-value of Function A when [tex]$x = 10$[/tex]
Using the equation [tex]$y = x - 2$[/tex], substitute [tex]$x = 10$[/tex]:
[tex]\[ y = 10 - 2 \][/tex]
[tex]\[ y = 8 \][/tex]
So, the [tex]$y$[/tex]-value of Function A when [tex]$x = 10$[/tex] is 8.
### Step 3: Calculate the y-value of Function B when [tex]$x = 10$[/tex]
We are given the equation for Function B as:
[tex]\[ y = \frac{1}{2}x - 3 \][/tex]
Substitute [tex]$x = 10$[/tex] into this equation:
[tex]\[ y = \frac{1}{2}(10) - 3 \][/tex]
[tex]\[ y = 5 - 3 \][/tex]
[tex]\[ y = 2 \][/tex]
So, the [tex]$y$[/tex]-value of Function B when [tex]$x = 10$[/tex] is 2.
### Step 4: Compare the y-values of Function A and Function B when [tex]$x = 10$[/tex]
- The [tex]$y$[/tex]-value of Function A when [tex]$x = 10$[/tex] is 8.
- The [tex]$y$[/tex]-value of Function B when [tex]$x = 10$[/tex] is 2.
Since 8 is greater than 2, the statement that is true is:
The [tex]$y$[/tex]-value of Function A when [tex]$x = 10$[/tex] is greater than the [tex]$y$[/tex]-value of Function B when [tex]$x = 10$[/tex].
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