Certainly! Let's solve the given system of equations using the elimination method step by step:
1. Given Equations:
[tex]\[
x + 3y = 17 \quad \text{(1)}
\][/tex]
[tex]\[
x + 2y = 14 \quad \text{(2)}
\][/tex]
2. Subtract Equation (2) from Equation (1) to eliminate [tex]\( x \)[/tex]:
[tex]\[
(x + 3y) - (x + 2y) = 17 - 14
\][/tex]
Simplifying the left side:
[tex]\[
x + 3y - x - 2y = 3
\][/tex]
This gives:
[tex]\[
y = 3
\][/tex]
3. Substitute [tex]\( y = 3 \)[/tex] back into Equation (2) to find [tex]\( x \)[/tex]:
[tex]\[
x + 2(3) = 14
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x + 6 = 14
\][/tex]
[tex]\[
x = 14 - 6
\][/tex]
[tex]\[
x = 8
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
x = 8, \quad y = 3
\][/tex]
---
Now, we will solve the inequality and represent it on a number line.
Example Inequality: Let's solve [tex]\( 3x - 7 \leq 2x + 5 \)[/tex].
4. Solve the Inequality:
[tex]\[
3x - 7 \leq 2x + 5
\][/tex]
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[
3x - 2x - 7 \leq 5
\][/tex]
Simplify:
[tex]\[
x - 7 \leq 5
\][/tex]
Add 7 to both sides:
[tex]\[
x \leq 12
\][/tex]
5. Represent the Solution on a Number Line:
To represent [tex]\( x \leq 12 \)[/tex] on a number line, you draw a line, mark the point 12, and shade everything to the left of 12, including the point itself (using a closed circle to include 12).
---
```
Number Line Representation:
--------------------------------------
-∞ ... -3 -2 -1 0 1 2 3 ...... 10 11 [12] -> all values
including 12
--------------------------------------
```
Here, "[" denotes a closed interval indicating [tex]\( x \leq 12 \)[/tex].
Thus, [tex]\( x \leq 12 \)[/tex] means any value of [tex]\( x \)[/tex] that is less than or equal to 12.
