Question

complete the equation that represents the relationship between c and w.

c	w
6	25
7	26
8	27

w = \boxed{}

Explanation:

Step1: Analyze the pattern

Looking at the table, when \( c = 5 \), \( w = 24 \); \( c = 6 \), \( w = 25 \); \( c = 7 \), \( w = 26 \); \( c = 8 \), \( w = 27 \). We can see that \( w \) is always \( c + 19 \)? Wait, no, let's check the difference. \( 24 - 5 = 19 \), \( 25 - 6 = 19 \), \( 26 - 7 = 19 \), \( 27 - 8 = 19 \)? Wait, no, \( 5 + 19 = 24 \), \( 6 + 19 = 25 \), yes! Wait, or maybe \( w = c + 19 \)? Wait, no, \( 5 + 19 = 24 \), \( 6 + 19 = 25 \), \( 7 + 19 = 26 \), \( 8 + 19 = 27 \). Wait, but another way: when \( c \) increases by 1, \( w \) increases by 1. So the slope (rate of change) is \( \frac{\Delta w}{\Delta c}=\frac{25 - 24}{6 - 5}=1 \). So it's a linear relationship. Let's use the point - slope form. Let's take the point \( (c = 5, w = 24) \). The equation of a line is \( w - w_1=m(c - c_1) \), where \( m = 1 \), \( c_1 = 5 \), \( w_1 = 24 \). So \( w - 24=1\times(c - 5) \), which simplifies to \( w=c - 5 + 24 \), so \( w=c + 19 \)? Wait, no, \( c - 5+24=c + 19 \). Wait, but let's check with \( c = 6 \): \( 6 + 19 = 25 \), which matches. \( c = 7 \): \( 7+19 = 26 \), matches. \( c = 8 \): \( 8 + 19 = 27 \), matches. Alternatively, we can see that \( w=c + 19 \)? Wait, no, wait \( 24-5 = 19 \), \( 25 - 6 = 19 \), so \( w=c + 19 \). Wait, but another way: when \( c = 5 \), \( w = 24 \), so \( w=c + 19 \) (since \( 5+19 = 24 \)). Or, let's see the difference between \( w \) and \( c \): \( 24-5 = 19 \), \( 25 - 6 = 19 \), so \( w=c + 19 \). Wait, but also, since the change in \( w \) is 1 when change in \( c \) is 1, the slope \( m = 1 \), and using the equation \( w=mc + b \). Let's plug in \( c = 5 \), \( w = 24 \): \( 24=1\times5 + b \), so \( b = 24 - 5=19 \). So the equation is \( w=c + 19 \)? Wait, no, wait \( 5 + 19 = 24 \), \( 6+19 = 25 \), yes. Wait, but wait, maybe I made a mistake. Wait, \( 24-5 = 19 \), \( 25 - 6 = 19 \), so \( w=c + 19 \). Alternatively, \( w=c + 19 \). Wait, but let's check again. If \( c = 5 \), \( w = 24 \): \( 5+19 = 24 \). \( c = 6 \), \( 6 + 19=25 \). Correct. So the equation is \( w=c + 19 \)? Wait, no, wait, maybe it's \( w=c + 19 \)? Wait, but another way: when \( c \) is 5, \( w \) is 24, so \( w=c + 19 \). Yes, that seems to fit.

Wait, wait, no, wait a second. Let's do the slope - intercept form properly. The general form of a linear equation is \( w=mc + b \), where \( m \) is the slope and \( b \) is the y - intercept.

To find \( m \), we use two points. Let's take \( (c_1,w_1)=(5,24) \) and \( (c_2,w_2)=(6,25) \).

The slope \( m=\frac{w_2 - w_1}{c_2 - c_1}=\frac{25 - 24}{6 - 5}=\frac{1}{1}=1 \).

Now, substitute \( m = 1 \), \( c = 5 \), and \( w = 24 \) into \( w=mc + b \):

\( 24=1\times5 + b \)

\( 24 = 5 + b \)

Subtract 5 from both sides: \( b=24 - 5 = 19 \).

So the equation is \( w=1\times c+19 \), which simplifies to \( w=c + 19 \).

Wait, but let's check with \( c = 7 \): \( w=7 + 19 = 26 \), which matches the table. \( c = 8 \): \( w=8 + 19 = 27 \), which also matches. So the equation is \( w=c + 19 \). Wait, but that seems odd. Wait, maybe I misread the table. Let me check again. The table has \( c \) values 5,6,7,8 and \( w \) values 24,25,26,27. So when \( c \) increases by 1, \( w \) increases by 1. So the relationship is \( w=c + 19 \)? Wait, no, \( 5+19 = 24 \), \( 6 + 19=25 \), yes. So the equation is \( w=c + 19 \).

Answer:

\( w=c + 19 \)