Question

we used substitution as a method to solve a system of equations last time but there is one last method that can be used to solve a system of equations.
adding two equations together is an example of a linear combination. explore the following linear combinations.

1. \\(\

$$\begin{array}{r} 5x + y = 7 \\\\ +\\quad 2x - y = 14 \\end{array}$$

\\)

1. \\(\

$$\begin{array}{r} x + 3y = 4 \\\\ +\\quad -x + y = 8 \\end{array}$$

\\)

1. \\(\

$$\begin{array}{r} 3x - 2y = 6 \\\\ +\\quad -x + 2y = 2 \\end{array}$$

\\)

diy: solve the following systems by elimination.

1. \\(\

$$\begin{array}{r} x - 3y = -7 \\\\ +\\quad 2x + 3y = 13 \\end{array}$$

\\)

1. \\(\

$$\begin{array}{r} 3x - y = -10 \\\\ +\\quad 5x + y = 2 \\end{array}$$

\\)

Explanation:

Explore linear combination 1

Add the two equations:

$$
LATEXBLOCK0
$$

Substitute \(x = 3\) into the first equation:

$$
LATEXBLOCK1
$$

Explore linear combination 2

Add the two equations:

$$
LATEXBLOCK2
$$

Substitute \(y = 3\) into the first equation:

$$
LATEXBLOCK3
$$

Explore linear combination 3

Add the two equations:

$$
LATEXBLOCK4
$$

Substitute \(x = 4\) into the first equation:

$$
LATEXBLOCK5
$$

Solve DIY system 4

Add the two equations:

$$
LATEXBLOCK6
$$

Substitute \(x = 2\) into the first equation:

$$
LATEXBLOCK7
$$

Solve DIY system 5

Add the two equations:

$$
LATEXBLOCK8
$$

Substitute \(x = -1\) into the second equation:

$$
LATEXBLOCK9
$$

Answer:

Question 1

$$(x, y) = (3, -8)$$

Question 2

$$(x, y) = (-5, 3)$$

Question 3

$$(x, y) = (4, 3)$$

Question 4

$$(x, y) = (2, 3)$$

Question 5

$$(x, y) = (-1, 7)$$