Find a and b if the graph ax+2y=b passes through (2,5) and is parallel to 3x-5y=7
i find a= -6/5 b= 38/5
is it correct? thanks
Your answer is correct.
Equations in slope intercept form:
a x + 2 y = b
y = ( - a x + b ) / 2
y = ( - a / 2 ) x + b / 2
3 x - 5 y = 7 -> y = ( - 3 x + 7 ) / - 5
y = ( 3 / 5 ) x - 7 / 5
Parallel lines have the same slope.
( - a / 2 ) = 3 / 5
Cross multiply:
- 5 a = 6
a = - 6 / 5
a x + 2 y = b passes through ( 2 , 5 ) mean x = 2 , y = 5
- ( 6 / 5 ) x + 2 y = b
- ( 6 / 5 ) ∙ 2 + 2 ∙ 5 = b
- 12 / 5 + 10 = b
- 12 / 5 + 50 / 5 = b
38 / 5 = b
b = 38 / 5
a x + 2 y = b
- ( 6 / 5 ) x + 2 y = 38 / 5
Well, your calculations seem a bit off, but don't worry, I'm here to help you out with a smile! Let's work through this together.
To find the values of a and b, we can start by rearranging the equation 3x - 5y = 7 into slope-intercept form y = mx + b. When we do this, we get:
-5y = -3x + 7
y = (3/5)x - 7/5
Since the given line is parallel to this line, it will also have a slope of 3/5.
Now, let's plug in the coordinates (2,5) into the equation ax + 2y = b and solve for a and b.
Using the coordinates (2,5), we have:
a(2) + 2(5) = b
2a + 10 = b
Since we want the values of a and b that satisfy the given conditions, we can substitute the slope 3/5 into the equation:
2a + 10 = (3/5)(2) + (3/5)(10)
2a + 10 = 6/5 + 30/5
2a + 10 = 36/5
Now, let's solve for a:
2a = (36/5) - (10)
2a = 36/5 - 50/5
2a = -14/5
a = -7/5
So, we have found that a = -7/5.
Now, let's substitute this value into the equation to find b:
2a + 10 = b
2(-7/5) + 10 = b
-14/5 + 10 = b
-14/5 + 50/5 = b
36/5 = b
So, we have found that b = 36/5.
Therefore, the correct values of a and b are a = -7/5 and b = 36/5.
Keep smiling and stay positive!
To find the values of a and b, we need to use two pieces of information:
1. The graph of ax + 2y = b passes through the point (2, 5).
2. The graph is parallel to the line 3x - 5y = 7.
Let's start by finding the slope of the line 3x - 5y = 7. We can do this by converting the equation to slope-intercept form (y = mx + b):
3x - 5y = 7
-5y = -3x + 7
y = (3/5)x - 7/5
The slope of the line is 3/5. Since the line ax + 2y = b is parallel to this line, it will have the same slope. Therefore, we can rewrite ax + 2y = b as:
ax + 2y = b
2y = -ax + b
y = (-a/2)x + b/2
Now, we substitute the coordinates of the point (2, 5) into the equation to get:
5 = (-a/2)(2) + b/2
5 = -a + b/2
10 = -2a + b
From here, we have a system of equations:
(1) 10 = -2a + b
(2) y = (-a/2)x + b/2
To solve for a and b, we can use the fact that the graph passes through (2, 5). Substituting x = 2 and y = 5 into equation (2), we get:
5 = (-a/2)(2) + b/2
5 = -a + b/2
Now we can substitute this equation into equation (1):
10 = -2a + (-a + b/2)
10 = -2a - a + b/2
10 = -3a + b/2
20 = -6a + b
From equations (1) and (3):
-2a + b = 10
-6a + b = 20
We can solve this system of equations using various methods, such as substitution or elimination. Let's use the elimination method to cancel out the b terms:
-2a + b - (-6a + b) = 10 - 20
-2a + b + 6a - b = -10
4a = -10
a = -10/4
a = -5/2
Substituting a = -5/2 into equation (1):
-2(-5/2) + b = 10
5 + b = 10
b = 10 - 5
b = 5
Therefore, the values of a and b are a = -5/2 and b = 5. Your answer, a = -6/5 and b = 38/5, is incorrect.
To find the values of a and b, we'll use the fact that the given graph passes through the point (2,5) and is parallel to the line 3x - 5y = 7.
First, let's find the slope of the given line by converting it to slope-intercept form (y = mx + b):
3x - 5y = 7
-5y = -3x + 7
y = (3/5)x - 7/5
We know that the graph we're looking for is parallel to this line, meaning it must have the same slope. Therefore, the slope of the desired line is also 3/5.
Now, we can use the point-slope form of a linear equation:
y - y₁ = m(x - x₁)
where (x₁, y₁) is the point (2, 5), and m is the slope (3/5).
Substituting the values:
y - 5 = (3/5)(x - 2)
Expanding:
y - 5 = (3/5)x - 6/5
Isolating y:
y = (3/5)x - 6/5 + 5
y = (3/5)x - 6/5 + 25/5
y = (3/5)x + 19/5
Comparing the equation of the desired line with the standard form of a line, ax + 2y = b, we can see that a = 3/5, and b = 19/5.
Therefore, a = 3/5 and b = 19/5.
It seems there was an error in the calculation you provided. The correct values for a and b are a = 3/5 and b = 19/5, not a = -6/5 and b = 38/5.