Question

given a ratio
find b on \\(\overrightarrow{ac}\\) such that the ratio of ab to bc is 3:4.
use the section formula to determine the coordinate of point b.
\\( b = \frac{nx_1 + mx_2}{m + n} \\)
section formula
\\( m = 3, n = 4, x_1 = -5, \\) and \\( x_2 = \\) partially visible
\\( = \frac{\\________}{\\________} = \frac{1}{7} \\) partial
so, b is located at \\____ on the number line.
includes number line diagrams with points a, b, c at coordinates -5, (unknown), 7 and another diagram with a at -5, b at 0, c at 7 (partial)

Explanation:

Step1: Identify given values

We know \( m = 3 \), \( n = 4 \), \( x_1=-5 \) (coordinate of \( A \)), and \( x_2 = 7 \) (coordinate of \( C \)) from the number line. The section formula is \( B=\frac{nx_1+mx_2}{m + n} \).

Step2: Substitute values into formula

Substitute \( m = 3 \), \( n = 4 \), \( x_1=-5 \), and \( x_2 = 7 \) into the formula:
\[ B=\frac{4\times(-5)+3\times7}{3 + 4} \]

Step3: Calculate numerator and denominator

First, calculate the numerator: \( 4\times(-5)+3\times7=-20 + 21 = 1 \).
The denominator is \( 3 + 4=7 \).

Step4: Find the value of \( B \)

So, \( B=\frac{1}{7} \)? Wait, no, wait. Wait, maybe I misread \( x_2 \). Wait, looking at the number line, \( C \) is at 7? Wait, no, in the lower number line, \( C \) is at 7? Wait, no, the upper number line: \( A \) is at -5, \( C \) is at 7. Wait, but let's recheck. Wait, the section formula: when a point \( B \) divides \( \overline{AC} \) in the ratio \( AB:BC=m:n \), then the coordinate of \( B \) is \( \frac{nx_1+mx_2}{m + n} \). Wait, \( AB:BC = 3:4 \), so \( m = 3 \), \( n = 4 \), so \( x_1 \) is \( A \) (-5), \( x_2 \) is \( C \) (7). So substituting:

Numerator: \( 4\times(-5)+3\times7=-20 + 21 = 1 \)

Denominator: \( 3 + 4 = 7 \)

Wait, but in the lower diagram, \( B \) is at 0? Wait, maybe I made a mistake. Wait, no, let's check again. Wait, maybe \( x_2 \) is 7? Wait, no, in the upper number line, \( C \) is at 7, \( A \) at -5. So the length of \( AC \) is \( 7 - (-5)=12 \). The ratio \( AB:BC = 3:4 \), so total parts \( 3 + 4 = 7 \). Each part is \( \frac{12}{7} \). Then \( AB = 3\times\frac{12}{7}=\frac{36}{7}\approx5.14 \), so \( B \) is at \( -5+\frac{36}{7}=\frac{-35 + 36}{7}=\frac{1}{7}\approx0.14 \). But in the lower diagram, \( B \) is at 0? Wait, maybe the \( x_2 \) is 7? Wait, maybe the problem has \( x_2 = 7 \). So according to the formula, \( B=\frac{4\times(-5)+3\times7}{7}=\frac{-20 + 21}{7}=\frac{1}{7} \)? But the lower diagram shows \( B \) at 0. Wait, maybe I misread the ratio. Wait, the ratio is \( AB:BC = 3:4 \), so \( m = 3 \), \( n = 4 \), so the formula is correct. Wait, maybe the \( x_2 \) is 7. So the calculation is \( \frac{4\times(-5)+3\times7}{3 + 4}=\frac{-20 + 21}{7}=\frac{1}{7} \). But the lower diagram has \( B \) at 0. Maybe there's a typo, but according to the given formula and values, let's proceed.

Wait, maybe the \( x_2 \) is 7, so:

\( B=\frac{4\times(-5)+3\times7}{3 + 4}=\frac{-20 + 21}{7}=\frac{1}{7} \). But the lower diagram shows \( B \) at 0. Maybe the problem intended \( x_2 = 5 \) or something else. Wait, no, the upper number line: \( A \) at -5, \( C \) at 7. So the correct calculation is \( \frac{4\times(-5)+3\times7}{7}=\frac{1}{7} \). But maybe the problem has a mistake, but according to the given formula and values, let's do the calculation.

Wait, maybe I mixed up \( m \) and \( n \). The ratio \( AB:BC = 3:4 \), so \( AB = 3k \), \( BC = 4k \), so the point \( B \) is closer to \( A \) or \( C \)? Since \( 3:4 \), \( AB \) is 3 parts, \( BC \) is 4 parts, so \( B \) is closer to \( A \). Wait, the section formula: if \( B \) divides \( AC \) internally in the ratio \( m:n \) ( \( AB:BC = m:n \) ), then the coordinate is \( \frac{nx_1+mx_2}{m + n} \). So \( m = 3 \), \( n = 4 \), \( x_1=-5 \), \( x_2 = 7 \). So:

\( nx_1 + mx_2 = 4\times(-5)+3\times7=-20 + 21 = 1 \)

\( m + n = 7 \)

So \( B=\frac{1}{7} \). But the lower diagram shows \( B \) at 0. Maybe the \( x_2 \) is 5? Wait, no, the upper number line has \( C \) at 7. Maybe the problem is correct, and the diagram is just a sketch. So according to the calculation, \( B \) is at \( \frac{1}{7} \)? Wait, no, wait, maybe I made a mistake in the formula. Wait, the section formula for a number line (1 - D) is: if a point divides the segment between \( x_1 \) and \( x_2 \) in the ratio \( k:1 \), but here the ratio is \( AB:BC = 3:4 \), so the ratio of \( AB \) to \( BC \) is 3:4, so the ratio of \( AB \) to \( AC \) is 3:7, so the coordinate of \( B \) is \( x_1+\frac{3}{7}(x_2 - x_1) \). Let's calculate that: \( -5+\frac{3}{7}(7 - (-5))=-5+\frac{3}{7}(12)=-5+\frac{36}{7}=\frac{-35 + 36}{7}=\frac{1}{7} \). So that's correct. So the coordinate of \( B \) is \( \frac{1}{7} \). But the lower diagram shows \( B \) at 0, maybe it's a rough sketch. So according to the calculation, the answer is \( \frac{1}{7} \).

Wait, but the problem says "Use the Section Formula to determine the coordinate of point B. \( B=\frac{nx_1+mx_2}{m + n} \), \( m = 3 \), \( n = 4 \), \( x_1=-5 \), and \( x_2 = 7 \)". So substituting:

\( B=\frac{4\times(-5)+3\times7}{3 + 4}=\frac{-20 + 21}{7}=\frac{1}{7} \). So that's the calculation.

Answer:

\(\frac{1}{7}\)