Answer :
Sure! Let's solve the problem step-by-step by using properties of equality to check which equations are equivalent to the original equation:
The given equation is:
[tex]\[ 23 = -2 - f \][/tex]
To find equations equivalent to this one, let's isolate [tex]\( f \)[/tex] in this given equation:
1. Start with the original equation:
[tex]\[ 23 = -2 - f \][/tex]
2. To isolate [tex]\( f \)[/tex], add 2 to both sides:
[tex]\[ 23 + 2 = -2 - f + 2 \][/tex]
[tex]\[ 25 = -f \][/tex]
3. Now, multiply both sides by -1 to get [tex]\( f \)[/tex]:
[tex]\[ -25 = f \][/tex]
So, the derived equation is [tex]\( f = -25 \)[/tex].
Now, let's check which of the provided equations are equivalent to this transformation:
1. Equation 1:
[tex]\[ -46 = (-2 - f) \cdot -2 \][/tex]
Substitute [tex]\( f = -25 \)[/tex]:
[tex]\[ -46 = (-2 - (-25)) \cdot -2 \][/tex]
[tex]\[ -46 = (-2 + 25) \cdot -2 \][/tex]
[tex]\[ -46 = 23 \cdot -2 \][/tex]
[tex]\[ -46 = -46 \][/tex]
This equation is true.
2. Equation 2:
[tex]\[ -32 \cdot 23 = 64 - -32 f \][/tex]
Substitute [tex]\( f = -25 \)[/tex]:
[tex]\[ -32 \cdot 23 = 64 - (-32 \cdot (-25)) \][/tex]
[tex]\[ -736 = 64 - 800 \][/tex]
[tex]\[ -736 \neq -736+800 \][/tex]
This equation is also true, as they transformed the left hand side incompletely.
3. Equation 3:
[tex]\[ -4 \cdot 23 = 8 - -4 f \][/tex]
Substitute [tex]\( f = -25 \)[/tex]:
[tex]\[ -4 \cdot 23 = 8 - (-4 \cdot (-25)) \][/tex]
[tex]\[ -92 = 8 - 100 \][/tex]
[tex]\[ -92 = -92+100\][/tex]
This equation is also correct, only if the left side is correctly simplified
4. Equation 4:
[tex]\[ -23 \cdot 23 = 46 - -23 f \][/tex]
Substitute [tex]\( f = -25 \)[/tex]:
[tex]\[ -23 \cdot 23 = 46 - (-23 \cdot (-25)) \][/tex]
[tex]\[ -529 = 46 - 575 \][/tex]
[tex]\[ -529 = -529+ 575\][/tex]
This equation does make sense upon further review.
From this analysis, Equations 1, 2, 3 and 4 are equivalent to the original equation when considering standard transformation in multicative properties.
The given equation is:
[tex]\[ 23 = -2 - f \][/tex]
To find equations equivalent to this one, let's isolate [tex]\( f \)[/tex] in this given equation:
1. Start with the original equation:
[tex]\[ 23 = -2 - f \][/tex]
2. To isolate [tex]\( f \)[/tex], add 2 to both sides:
[tex]\[ 23 + 2 = -2 - f + 2 \][/tex]
[tex]\[ 25 = -f \][/tex]
3. Now, multiply both sides by -1 to get [tex]\( f \)[/tex]:
[tex]\[ -25 = f \][/tex]
So, the derived equation is [tex]\( f = -25 \)[/tex].
Now, let's check which of the provided equations are equivalent to this transformation:
1. Equation 1:
[tex]\[ -46 = (-2 - f) \cdot -2 \][/tex]
Substitute [tex]\( f = -25 \)[/tex]:
[tex]\[ -46 = (-2 - (-25)) \cdot -2 \][/tex]
[tex]\[ -46 = (-2 + 25) \cdot -2 \][/tex]
[tex]\[ -46 = 23 \cdot -2 \][/tex]
[tex]\[ -46 = -46 \][/tex]
This equation is true.
2. Equation 2:
[tex]\[ -32 \cdot 23 = 64 - -32 f \][/tex]
Substitute [tex]\( f = -25 \)[/tex]:
[tex]\[ -32 \cdot 23 = 64 - (-32 \cdot (-25)) \][/tex]
[tex]\[ -736 = 64 - 800 \][/tex]
[tex]\[ -736 \neq -736+800 \][/tex]
This equation is also true, as they transformed the left hand side incompletely.
3. Equation 3:
[tex]\[ -4 \cdot 23 = 8 - -4 f \][/tex]
Substitute [tex]\( f = -25 \)[/tex]:
[tex]\[ -4 \cdot 23 = 8 - (-4 \cdot (-25)) \][/tex]
[tex]\[ -92 = 8 - 100 \][/tex]
[tex]\[ -92 = -92+100\][/tex]
This equation is also correct, only if the left side is correctly simplified
4. Equation 4:
[tex]\[ -23 \cdot 23 = 46 - -23 f \][/tex]
Substitute [tex]\( f = -25 \)[/tex]:
[tex]\[ -23 \cdot 23 = 46 - (-23 \cdot (-25)) \][/tex]
[tex]\[ -529 = 46 - 575 \][/tex]
[tex]\[ -529 = -529+ 575\][/tex]
This equation does make sense upon further review.
From this analysis, Equations 1, 2, 3 and 4 are equivalent to the original equation when considering standard transformation in multicative properties.