Answer :
To find an equation that describes the relationship between \( a \) and \( b \) given the points \((8, 3)\), \((24, 9)\), and \((40, 15)\), we can assume that the relationship is linear. This means the equation can be written in the form:
[tex]\[ b = ma + c \][/tex]
where \( m \) is the slope of the line, and \( c \) is the y-intercept. Let's determine these values step by step.
### Step 1: Calculate the slope (\( m \))
The slope of a line that passes through two points \((a_1, b_1)\) and \((a_2, b_2)\) is given by:
[tex]\[ m = \frac{b_2 - b_1}{a_2 - a_1} \][/tex]
We can use any pair of the given points to calculate the slope. Let's use the points \((8, 3)\) and \((24, 9)\):
[tex]\[ m = \frac{9 - 3}{24 - 8} = \frac{6}{16} = \frac{3}{8} \][/tex]
Thus, the slope \( m = \frac{3}{8} \).
### Step 2: Determine the y-intercept (\( c \))
To find the y-intercept \( c \), we can use one of the given points along with the slope we just calculated. Let's use the point \((8, 3)\). Substitute \( a = 8 \), \( b = 3 \), and \( m = \frac{3}{8} \) into the equation \( b = ma + c \):
[tex]\[ 3 = \frac{3}{8}(8) + c \][/tex]
[tex]\[ 3 = 3 + c \][/tex]
Subtract 3 from both sides to solve for \( c \):
[tex]\[ 3 - 3 = c \][/tex]
[tex]\[ c = 0 \][/tex]
### Step 3: Write the final equation
Now that we have \( m = \frac{3}{8} \) and \( c = 0 \), we can write the equation describing the relationship between \( a \) and \( b \):
[tex]\[ b = \frac{3}{8}a \][/tex]
### Verification
To ensure that our equation is correct, we can verify it with the other points \((24, 9)\) and \((40, 15)\):
1. For \((24, 9)\):
[tex]\[ b = \frac{3}{8}(24) = 9 \][/tex] (Correct)
2. For \((40, 15)\):
[tex]\[ b = \frac{3}{8}(40) = 15 \][/tex] (Correct)
Thus, the linear equation that correctly describes the relationship between \( a \) and \( b \) is:
[tex]\[ b = \frac{3}{8}a \][/tex]
[tex]\[ b = ma + c \][/tex]
where \( m \) is the slope of the line, and \( c \) is the y-intercept. Let's determine these values step by step.
### Step 1: Calculate the slope (\( m \))
The slope of a line that passes through two points \((a_1, b_1)\) and \((a_2, b_2)\) is given by:
[tex]\[ m = \frac{b_2 - b_1}{a_2 - a_1} \][/tex]
We can use any pair of the given points to calculate the slope. Let's use the points \((8, 3)\) and \((24, 9)\):
[tex]\[ m = \frac{9 - 3}{24 - 8} = \frac{6}{16} = \frac{3}{8} \][/tex]
Thus, the slope \( m = \frac{3}{8} \).
### Step 2: Determine the y-intercept (\( c \))
To find the y-intercept \( c \), we can use one of the given points along with the slope we just calculated. Let's use the point \((8, 3)\). Substitute \( a = 8 \), \( b = 3 \), and \( m = \frac{3}{8} \) into the equation \( b = ma + c \):
[tex]\[ 3 = \frac{3}{8}(8) + c \][/tex]
[tex]\[ 3 = 3 + c \][/tex]
Subtract 3 from both sides to solve for \( c \):
[tex]\[ 3 - 3 = c \][/tex]
[tex]\[ c = 0 \][/tex]
### Step 3: Write the final equation
Now that we have \( m = \frac{3}{8} \) and \( c = 0 \), we can write the equation describing the relationship between \( a \) and \( b \):
[tex]\[ b = \frac{3}{8}a \][/tex]
### Verification
To ensure that our equation is correct, we can verify it with the other points \((24, 9)\) and \((40, 15)\):
1. For \((24, 9)\):
[tex]\[ b = \frac{3}{8}(24) = 9 \][/tex] (Correct)
2. For \((40, 15)\):
[tex]\[ b = \frac{3}{8}(40) = 15 \][/tex] (Correct)
Thus, the linear equation that correctly describes the relationship between \( a \) and \( b \) is:
[tex]\[ b = \frac{3}{8}a \][/tex]
