To find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] such that the midpoint between [tex]\((x, 1)\)[/tex] and [tex]\((-3, y)\)[/tex] is [tex]\((2, 1)\)[/tex], we can use the midpoint formula. The midpoint formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Given the points [tex]\((x, 1)\)[/tex] and [tex]\((-3, y)\)[/tex], and knowing that their midpoint is [tex]\((2, 1)\)[/tex], we can set up two equations by comparing the coordinates of the midpoint.
### Step-by-Step Solution
1. Determine the x-coordinate of the midpoint:
The x-coordinate of the midpoint is found by averaging the x-coordinates of the given points:
[tex]\[
\frac{x + (-3)}{2} = 2
\][/tex]
Simplify the equation:
[tex]\[
\frac{x - 3}{2} = 2
\][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[
x - 3 = 4
\][/tex]
Add 3 to both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 7
\][/tex]
2. Determine the y-coordinate of the midpoint:
The y-coordinate of the midpoint is found by averaging the y-coordinates of the given points:
[tex]\[
\frac{1 + y}{2} = 1
\][/tex]
Simplify the equation:
[tex]\[
\frac{1 + y}{2} = 1
\][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[
1 + y = 2
\][/tex]
Subtract 1 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 1
\][/tex]
### Conclusion
The values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the condition that the midpoint between the points [tex]\((x, 1)\)[/tex] and [tex]\((-3, y)\)[/tex] is [tex]\((2, 1)\)[/tex] are:
[tex]\[
x = 7 \quad \text{and} \quad y = 1
\][/tex]
