Certainly! Let's analyze whether the points [tex]\((1, 5)\)[/tex] and [tex]\((0, 0)\)[/tex] lie on the given lines [tex]\(y = 2x + 3\)[/tex] and [tex]\(y = 3x\)[/tex].
### Step 1: Checking the point [tex]\((1, 5)\)[/tex]
#### For the equation [tex]\(y = 2x + 3\)[/tex]:
- Substitute [tex]\(x = 1\)[/tex] into the equation:
[tex]\[
y = 2(1) + 3 = 2 + 3 = 5
\][/tex]
- The resulting [tex]\(y\)[/tex] value is 5, which matches the [tex]\(y\)[/tex]-coordinate of the point. Thus, [tex]\((1, 5)\)[/tex] lies on the line [tex]\(y = 2x + 3\)[/tex].
#### For the equation [tex]\(y = 3x\)[/tex]:
- Substitute [tex]\(x = 1\)[/tex] into the equation:
[tex]\[
y = 3(1) = 3
\][/tex]
- The resulting [tex]\(y\)[/tex] value is 3, which does not match the [tex]\(y\)[/tex]-coordinate of the point. Thus, [tex]\((1, 5)\)[/tex] does not lie on the line [tex]\(y = 3x\)[/tex].
### Step 2: Checking the point [tex]\((0, 0)\)[/tex]
#### For the equation [tex]\(y = 2x + 3\)[/tex]:
- Substitute [tex]\(x = 0\)[/tex] into the equation:
[tex]\[
y = 2(0) + 3 = 0 + 3 = 3
\][/tex]
- The resulting [tex]\(y\)[/tex] value is 3, which does not match the [tex]\(y\)[/tex]-coordinate of the point. Thus, [tex]\((0, 0)\)[/tex] does not lie on the line [tex]\(y = 2x + 3\)[/tex].
#### For the equation [tex]\(y = 3x\)[/tex]:
- Substitute [tex]\(x = 0\)[/tex] into the equation:
[tex]\[
y = 3(0) = 0
\][/tex]
- The resulting [tex]\(y\)[/tex] value is 0, which matches the [tex]\(y\)[/tex]-coordinate of the point. Thus, [tex]\((0, 0)\)[/tex] lies on the line [tex]\(y = 3x\)[/tex].
### Conclusion
- The point [tex]\((1, 5)\)[/tex] lies on the line [tex]\(y = 2x + 3\)[/tex] but does not lie on the line [tex]\(y = 3x\)[/tex].
- The point [tex]\((0, 0)\)[/tex] does not lie on the line [tex]\(y = 2x + 3\)[/tex] but lies on the line [tex]\(y = 3x\)[/tex].
Therefore, the results can be summarized as:
[tex]\[
\begin{array}{cc}
& \begin{array}{cc} (1,5) & (0,0) \end{array} \\
\begin{array}{c} y = 2x + 3 \\ y = 3x \end{array} & \begin{array}{cc} [True, & False] \\ [False, & True] \end{array}
\end{array}
\][/tex]
