Question

The choices A–D below are a sequence of steps for solving the system 2x + y = 9 and x + 4y = 1 with the substitution method. Which step contains the first error?

Answer 1

Solution:

Given the system of equations:

$\begin{gathered} 2x+y=9\text{ --- equation 1} \\ x+4y=1\text{ ---- equation 2} \end{gathered}$

To solve,

Step 1: From equation 1, solve for y.

To solve for y, we make y the subject of the equation.

Thus,

$\begin{gathered} 2x+y=9 \\ add\text{ -2x to both sides of the equation,} \\ -2x+2x+y=-2x+9 \\ \Rightarrow y=-2x+9\text{ ----- equation 3} \end{gathered}$

Step 2: Substitute equation 3 into equation 2, to solve for x.

Thus,

$\begin{gathered} x+4y=1 \\ \Rightarrow x+4(-2x+9)=1 \\ open\text{ parentheses,} \\ x-8x+36=1 \\ \Rightarrow-7x+36=1 \\ subtract\text{ 36 from both sides of the equation,} \\ -7x+36-36=1-36 \\ \Rightarrow-7x=-35 \\ divide\text{ both sides by the coefficient of x, which is -7} \\ -\frac{7x}{-7}=-\frac{35}{-7} \\ \Rightarrow x=5 \end{gathered}$

Step 3: To solve for y, substitute the value of 5 for x into equation 3.

From equation 3,

$\begin{gathered} y=-2x+9 \\ where \\ x=5 \\ thus, \\ y=-2(5)+9=-10+9 \\ \Rightarrow y=-1 \end{gathered}$

Hence, the step that contains the first error is

$Substitute:The\text{ equation 2x+y=9 becomes 2x+\lparen-2x+9\rparen=9}$