Given that [tex]$h=\frac{20 k+3}{7}$[/tex], when [tex]k[/tex] is expressed in terms of [tex]h[/tex], [tex]k=\frac{a h+b}{c}[/tex]. Find the sum of [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex].

[tex][2 m][/tex]



Answer :

To solve for [tex]\( k \)[/tex] in terms of [tex]\( h \)[/tex] given the equation [tex]\( h = \frac{20k + 3}{7} \)[/tex], we need to isolate [tex]\( k \)[/tex]. Here’s a step-by-step solution:

1. Start with the given equation:
[tex]\[ h = \frac{20k + 3}{7} \][/tex]

2. Multiply both sides by 7 to eliminate the denominator:
[tex]\[ 7h = 20k + 3 \][/tex]

3. Isolate the term involving [tex]\( k \)[/tex] by subtracting 3 from both sides:
[tex]\[ 7h - 3 = 20k \][/tex]

4. Solve for [tex]\( k \)[/tex] by dividing both sides by 20:
[tex]\[ k = \frac{7h - 3}{20} \][/tex]

Now, we have expressed [tex]\( k \)[/tex] in terms of [tex]\( h \)[/tex] as:
[tex]\[ k = \frac{7h - 3}{20} \][/tex]

Comparing this to the given form [tex]\( k = \frac{ah + b}{c} \)[/tex], we can identify the constants [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ a = 7, \quad b = -3, \quad c = 20 \][/tex]

Finally, we need to find the sum of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ a + b + c = 7 + (-3) + 20 = 7 - 3 + 20 = 24 \][/tex]

Therefore, the sum of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is:
[tex]\[ \boxed{24} \][/tex]