Given [tex]\((x, 1)\)[/tex] and [tex]\((-3, y)\)[/tex], find [tex]\(x\)[/tex] and [tex]\(y\)[/tex] such that the midpoint between these two points is [tex]\((2,1)\)[/tex].



Answer :

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]

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