To identify the point of intersection between the given inequalities [tex]\( y > x - 1 \)[/tex] and [tex]\( y \leq 5x + 3 \)[/tex], follow these steps:
1. Convert inequalities to equations:
To find the precise point of intersection (where the two lines would cross if they were treated as equalities), let's first consider the equations corresponding to the inequalities:
- Equation 1: [tex]\( y = x - 1 \)[/tex]
- Equation 2: [tex]\( y = 5x + 3 \)[/tex]
2. Set the equations equal to each other:
To find the intersection point, solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] by setting the right-hand sides of the equations equal:
[tex]\[
x - 1 = 5x + 3
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Rearrange the equation to isolate [tex]\( x \)[/tex]:
[tex]\[
x - 1 - 5x = 3
\][/tex]
[tex]\[
-4x - 1 = 3
\][/tex]
[tex]\[
-4x = 4
\][/tex]
[tex]\[
x = -1
\][/tex]
4. Substitute [tex]\( x \)[/tex] back into either equation to find [tex]\( y \)[/tex]:
Using [tex]\( y = x - 1 \)[/tex]:
[tex]\[
y = -1 - 1
\][/tex]
[tex]\[
y = -2
\][/tex]
Alternatively, using [tex]\( y = 5x + 3 \)[/tex]:
[tex]\[
y = 5(-1) + 3
\][/tex]
[tex]\[
y = -5 + 3
\][/tex]
[tex]\[
y = -2
\][/tex]
The calculations confirm that the point [tex]\((-1, -2)\)[/tex] satisfies both equations.
5. Conclusion:
Therefore, the point of intersection between the inequalities [tex]\( y > x - 1 \)[/tex] and [tex]\( y \leq 5x + 3 \)[/tex] is [tex]\((-1, -2)\)[/tex].
