Question

summary

• substitution method for solving systems of equations: pick one of the two given equations and define one variable in terms of the other; substitute the expression for the defined variable into the other equation and solve; and substitute this solution back into either equation and solve for the remaining variable.
• combination method for solving systems of equations: eliminate one variable by combining the equations after multiplying either equation by a coefficient if necessary. after solving for one variable, substitute this value into either equation to solve for the remaining variable if required.

exercise 2
directions: choose the best answer from the four choices given. answers are on page 201.

1. if $x + y = 4$ and $y - x = 2$, then $x = $?

a. 0
b. 1
c. 2
d. 3

1. if $7x = 2$ and $3y - 7x = 10$, then $y = $?

a. 3
b. 4
c. 5
d. 6

1. if $\frac{x + y}{2} = 4$ and $x - y = 4$, then $x = $?

a. 2
b. 4
c. 6
d. 8

1. if $4x + 3y = 6$ and $3x + 2y = 5$, then $y = $?

a. -6
b. -2
c. 3
d. 6

1. if $x + y + z = 10$, $x - y - 2z = -13$, and $x + z = 6$, then $x = $?

a. 1
b. 3
c. 4
d. 5

Explanation:

Question 1

Step1: Define y from second equation

From \( y - x = 2 \), we get \( y = x + 2 \).

Step2: Substitute y into first equation

Substitute \( y = x + 2 \) into \( x + y = 4 \):
\( x + (x + 2) = 4 \)
\( 2x + 2 = 4 \)
\( 2x = 2 \)
\( x = 1 \)

Step1: Find x from first equation

From \( 7x = 2 \), \( x = \frac{2}{7} \).

Step2: Substitute x into second equation

Substitute \( x = \frac{2}{7} \) into \( 3y - 7x = 10 \):
\( 3y - 7(\frac{2}{7}) = 10 \)
\( 3y - 2 = 10 \)
\( 3y = 12 \)
\( y = 4 \)

Step1: Simplify first equation

From \( \frac{x + y}{2} = 4 \), multiply both sides by 2: \( x + y = 8 \).

Step2: Solve system with second equation

We have \( x + y = 8 \) and \( x - y = 4 \).
Add the two equations: \( (x + y) + (x - y) = 8 + 4 \)
\( 2x = 12 \)
\( x = 6 \)

Answer:

B. 1

Question 2