Math / Geometry

QuestionFind the value of $x$ and $y$ for each figure below.

Studdy Solution

STEP 1

1. We are given two triangles.
2. The sum of the angles in any triangle is $180^\circ$ .
3. We need to solve for $x$ and $y$ in each triangle using the given algebraic expressions for the angles.

STEP 2

1. Set up equations for the angles of the top triangle.
2. Solve the equations for the top triangle to find $x$ and $y$ .
3. Set up equations for the angles of the bottom triangle.
4. Solve the equations for the bottom triangle to find $x$ and $y$ .

STEP 3

For the top triangle, write the equation for the sum of the angles:
$98^\circ + (5x + 6)^\circ + (y - 9)^\circ = 180^\circ$

STEP 4

Simplify and solve the equation from Step 1:
$98 + 5x + 6 + y - 9 = 180$ $95 + 5x + y = 180$ $5x + y = 85$

STEP 5

We need another equation to solve for both $x$ and $y$ . However, we only have one equation for two variables. We need additional information or constraints to proceed. In this case, we assume the problem is designed such that the values of $x$ and $y$ are consistent across both triangles.

STEP 6

For the bottom triangle, write the equation for the sum of the angles:
$53^\circ + (3x + 8)^\circ + (8y + 10)^\circ = 180^\circ$

STEP 7

Simplify and solve the equation from Step 4:
$53 + 3x + 8 + 8y + 10 = 180$ $71 + 3x + 8y = 180$ $3x + 8y = 109$

STEP 8

Now we have a system of equations from both triangles:
1. $5x + y = 85$
2. $3x + 8y = 109$
Solve this system using substitution or elimination. Let's use substitution:
From equation 1, express $y$ in terms of $x$ :
$y = 85 - 5x$
Substitute $y$ in equation 2:
$3x + 8(85 - 5x) = 109$ $3x + 680 - 40x = 109$ $-37x + 680 = 109$ $-37x = 109 - 680$ $-37x = -571$ $x = \frac{571}{37}$ $x = 15.4324$ (rounded to four decimal places)
Substitute $x$ back into the expression for $y$ :
$y = 85 - 5(15.4324)$ $y = 85 - 77.162$ $y = 7.838$ (rounded to three decimal places)
The values of $x$ and $y$ are approximately:
$x \approx 15.4324$ $y \approx 7.838$

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