Certainly! Let's solve these questions step by step.
### Question 41
We are given the simultaneous equations:
[tex]\[ 3x - y = 5 \][/tex]
[tex]\[ x + y = 7 \][/tex]
We need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Step 1: Solve the second equation for [tex]\( y \)[/tex].
[tex]\[ x + y = 7 \][/tex]
[tex]\[ y = 7 - x \][/tex]
Step 2: Substitute the expression for [tex]\( y \)[/tex] into the first equation.
[tex]\[ 3x - (7 - x) = 5 \][/tex]
[tex]\[ 3x - 7 + x = 5 \][/tex]
[tex]\[ 4x - 7 = 5 \][/tex]
Step 3: Solve for [tex]\( x \)[/tex].
[tex]\[ 4x = 12 \][/tex]
[tex]\[ x = 3 \][/tex]
Step 4: Substitute [tex]\( x = 3 \)[/tex] back into the expression for [tex]\( y \)[/tex]
[tex]\[ y = 7 - x \][/tex]
[tex]\[ y = 7 - 3 \][/tex]
[tex]\[ y = 4 \][/tex]
So, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are:
[tex]\[ x = 3, \; y = 4 \][/tex]
Thus, the correct option is:
d) [tex]\((3, 4)\)[/tex]
### Question 42
We need to evaluate the expression:
[tex]\[ |2| + |3| - |-5| + |-4| - |-2| \][/tex]
Step 1: Evaluate the absolute values.
Absolute values convert any number to its non-negative form.
[tex]\[ |2| = 2 \][/tex]
[tex]\[ |3| = 3 \][/tex]
[tex]\[ |-5| = 5 \][/tex]
[tex]\[ |-4| = 4 \][/tex]
[tex]\[ |-2| = 2 \][/tex]
Step 2: Substitute these values into the expression.
[tex]\[ 2 + 3 - 5 + 4 - 2 \][/tex]
Step 3: Perform the addition and subtraction step by step.
[tex]\[ 2 + 3 = 5 \][/tex]
[tex]\[ 5 - 5 = 0 \][/tex]
[tex]\[ 0 + 4 = 4 \][/tex]
[tex]\[ 4 - 2 = 2 \][/tex]
Thus, the value of the expression is:
d) 2
Summary of the answers:
41. d) [tex]\((3, 4)\)[/tex]
42. d) 2
