Answer :
Sure, let's solve this step-by-step:
Fiona's equation is given by:
[tex]\[ y = \frac{2}{5} x - 5 \][/tex]
We need to transform Fiona's equation into standard form, which is typically written as [tex]\( Ax + By = C \)[/tex].
1. Start by moving all terms involving variables to one side. Here's Fiona's equation again:
[tex]\[ y = \frac{2}{5} x - 5 \][/tex]
2. Multiply every term by 5 to eliminate the fraction:
[tex]\[ 5y = 2x - 25 \][/tex]
3. Rearrange the terms to get all the variable terms on one side and the constant term on the other:
[tex]\[ 2x - 5y = 25 \][/tex]
Now, we compare this equation with the given options. To match the equations, we'll rewrite them in the same standard form [tex]\( Ax + By = C \)[/tex] and see which one is equivalent to [tex]\( 2x - 5y = 25 \)[/tex]:
Option 1:
[tex]\[ x - \frac{5}{4} y = \frac{25}{4} \][/tex]
To clear the fraction, multiply the entire equation by 4:
[tex]\[ 4x - 5y = 25 \][/tex]
This equation is not equivalent to [tex]\( 2x - 5y = 25 \)[/tex].
Option 2:
[tex]\[ x - \frac{5}{2} y = \frac{25}{4} \][/tex]
To clear the fraction, multiply the entire equation by 4:
[tex]\[ 4x - 10y = 25 \][/tex]
This equation is not equivalent to [tex]\( 2x - 5y = 25 \)[/tex].
Option 3:
[tex]\[ x - \frac{5}{4} y = \frac{25}{2} \][/tex]
To clear the fraction, multiply the entire equation by 4:
[tex]\[ 4x - 5y = 50 \][/tex]
This equation is not equivalent to [tex]\( 2x - 5y = 25 \)[/tex].
Option 4:
[tex]\[ x - \frac{5}{2} y = \frac{25}{2} \][/tex]
To clear the fraction, multiply the entire equation by 2:
[tex]\[ 2x - 5y = 25 \][/tex]
This equation matches exactly with our transformed equation [tex]\( 2x - 5y = 25 \)[/tex].
Therefore, the correct solution is:
[tex]\[ x - \frac{5}{2} y = \frac{25}{2} \][/tex]
Hence, Henry's equation is:
[tex]\[ \boxed{4} \][/tex]
Fiona's equation is given by:
[tex]\[ y = \frac{2}{5} x - 5 \][/tex]
We need to transform Fiona's equation into standard form, which is typically written as [tex]\( Ax + By = C \)[/tex].
1. Start by moving all terms involving variables to one side. Here's Fiona's equation again:
[tex]\[ y = \frac{2}{5} x - 5 \][/tex]
2. Multiply every term by 5 to eliminate the fraction:
[tex]\[ 5y = 2x - 25 \][/tex]
3. Rearrange the terms to get all the variable terms on one side and the constant term on the other:
[tex]\[ 2x - 5y = 25 \][/tex]
Now, we compare this equation with the given options. To match the equations, we'll rewrite them in the same standard form [tex]\( Ax + By = C \)[/tex] and see which one is equivalent to [tex]\( 2x - 5y = 25 \)[/tex]:
Option 1:
[tex]\[ x - \frac{5}{4} y = \frac{25}{4} \][/tex]
To clear the fraction, multiply the entire equation by 4:
[tex]\[ 4x - 5y = 25 \][/tex]
This equation is not equivalent to [tex]\( 2x - 5y = 25 \)[/tex].
Option 2:
[tex]\[ x - \frac{5}{2} y = \frac{25}{4} \][/tex]
To clear the fraction, multiply the entire equation by 4:
[tex]\[ 4x - 10y = 25 \][/tex]
This equation is not equivalent to [tex]\( 2x - 5y = 25 \)[/tex].
Option 3:
[tex]\[ x - \frac{5}{4} y = \frac{25}{2} \][/tex]
To clear the fraction, multiply the entire equation by 4:
[tex]\[ 4x - 5y = 50 \][/tex]
This equation is not equivalent to [tex]\( 2x - 5y = 25 \)[/tex].
Option 4:
[tex]\[ x - \frac{5}{2} y = \frac{25}{2} \][/tex]
To clear the fraction, multiply the entire equation by 2:
[tex]\[ 2x - 5y = 25 \][/tex]
This equation matches exactly with our transformed equation [tex]\( 2x - 5y = 25 \)[/tex].
Therefore, the correct solution is:
[tex]\[ x - \frac{5}{2} y = \frac{25}{2} \][/tex]
Hence, Henry's equation is:
[tex]\[ \boxed{4} \][/tex]
