Question

error analysis
aniya was solving a system using the substitution method. below is her work. find her error or errors and fix it.
aniyas work
( y = 2x + 3 )...(1)
( 2x - 5y = 7 )...(2)
put ( y = 2x + 3 ) into eq 2
( 2x - 5(2x + 3) = 7 )
( 2x - 10x + 15 = 7 )
( -8x = 7 - 15 )
( -8x = -8 )
( x = 1 )
put ( x = 1 ) into eq 1
( y = 2(1) + 3 )
( y = 2 + 3 )
( y = 5 )
your correction

Explanation:

Step1: Identify the error in distribution

When expanding \(-5(2x + 3)\), the distributive property \(a(b + c)=ab+ac\) is used. Here, \(a=-5\), \(b = 2x\), \(c = 3\), so it should be \(-5\times2x+(-5)\times3=-10x - 15\), not \(-10x + 15\).

Step2: Correct the expansion and solve for \(x\)

Starting from \(2x-5(2x + 3)=7\), expand correctly:
\(2x-10x - 15=7\)
Combine like terms: \(-8x-15 = 7\)
Add 15 to both sides: \(-8x=7 + 15\)
\(-8x=22\)
Divide both sides by \(-8\): \(x=\frac{22}{-8}=-\frac{11}{4}\)

Step3: Find \(y\) using \(x = -\frac{11}{4}\) in equation (1)

Substitute \(x = -\frac{11}{4}\) into \(y = 2x+3\):
\(y=2\times(-\frac{11}{4})+3\)
\(y=-\frac{11}{2}+3\)
\(y=-\frac{11}{2}+\frac{6}{2}=-\frac{5}{2}\)

Answer:

Error: In the step \(2x - 5(2x + 3)=7\) expansion, \(-5(2x + 3)\) was incorrectly expanded as \(-10x + 15\) (should be \(-10x - 15\)).
Correct solution: \(x = -\frac{11}{4}\), \(y = -\frac{5}{2}\)