Math / Algebra

QuestionSolve the following system of equations. $\begin{array}{l} x^{2}+y^{2}=16 \\ x+4 y=2 \end{array}$
What are ALL the solutions? Round to 3 decimal places. $\begin{array}{l} (\text { smaller } x \text { coordinate, } \mathrm{y} 1)=(\text { Number }), \text { Number }) \\ (\text { larger } \mathrm{x} \text { coordinate, } \mathrm{y} 2)=(\text { Number }) \end{array}$

Studdy Solution

STEP 1

1. The system of equations consists of a circle equation $x^2 + y^2 = 16$ and a linear equation $x + 4y = 2$ .
2. The solution involves finding the intersection points of the circle and the line.
3. The solutions should be rounded to three decimal places.

STEP 2

1. Solve the linear equation for one variable.
2. Substitute the expression from the linear equation into the circle equation.
3. Solve the resulting quadratic equation.
4. Find the corresponding $y$ values for each $x$ .
5. Round the solutions to three decimal places.

STEP 3

Solve the linear equation $x + 4y = 2$ for $x$ :
$x = 2 - 4y$

STEP 4

Substitute $x = 2 - 4y$ into the circle equation $x^2 + y^2 = 16$ :
$(2 - 4y)^2 + y^2 = 16$

STEP 5

Expand and simplify the equation:
$(2 - 4y)^2 = 4 - 16y + 16y^2$ $4 - 16y + 16y^2 + y^2 = 16$ $17y^2 - 16y + 4 = 16$

STEP 6

Move all terms to one side to form a quadratic equation:
$17y^2 - 16y + 4 - 16 = 0$ $17y^2 - 16y - 12 = 0$

STEP 7

Solve the quadratic equation $17y^2 - 16y - 12 = 0$ using the quadratic formula:
$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 17$ , $b = -16$ , and $c = -12$ .

STEP 8

Calculate the discriminant and solve for $y$ :
$b^2 - 4ac = (-16)^2 - 4 \times 17 \times (-12)$ $= 256 + 816$ $= 1072$
$y = \frac{16 \pm \sqrt{1072}}{34}$

STEP 9

Calculate the square root and find the values of $y$ :
$\sqrt{1072} \approx 32.740$
$y_1 = \frac{16 + 32.740}{34} \approx 1.448$ $y_2 = \frac{16 - 32.740}{34} \approx -0.491$

STEP 10

Substitute $y_1$ and $y_2$ back into $x = 2 - 4y$ to find corresponding $x$ values:
For $y_1 = 1.448$ :
$x_1 = 2 - 4(1.448)$ $x_1 = 2 - 5.792$ $x_1 = -3.792$
For $y_2 = -0.491$ :
$x_2 = 2 - 4(-0.491)$ $x_2 = 2 + 1.964$ $x_2 = 3.964$

STEP 11

Round the solutions to three decimal places:
Smaller $x$ coordinate solution: $(x_1, y_1) = (-3.792, 1.448)$
Larger $x$ coordinate solution: $(x_2, y_2) = (3.964, -0.491)$
The solutions are:
$(\text{smaller } x \text{ coordinate, } y_1) = (-3.792, 1.448)$ $(\text{larger } x \text{ coordinate, } y_2) = (3.964, -0.491)$

Was this helpful?

Studdy Solution

STEP 1

1. The system of equations consists of a circle equation $x^2 + y^2 = 16$ and a linear equation $x + 4y = 2$ .
2. The solution involves finding the intersection points of the circle and the line.
3. The solutions should be rounded to three decimal places.

STEP 2

1. Solve the linear equation for one variable.
2. Substitute the expression from the linear equation into the circle equation.
3. Solve the resulting quadratic equation.
4. Find the corresponding $y$ values for each $x$ .
5. Round the solutions to three decimal places.

STEP 3

Solve the linear equation $x + 4y = 2$ for $x$ :
$x = 2 - 4y$

STEP 4

Substitute $x = 2 - 4y$ into the circle equation $x^2 + y^2 = 16$ :
$(2 - 4y)^2 + y^2 = 16$

STEP 5

Expand and simplify the equation:
$(2 - 4y)^2 = 4 - 16y + 16y^2$ $4 - 16y + 16y^2 + y^2 = 16$ $17y^2 - 16y + 4 = 16$

STEP 6

Move all terms to one side to form a quadratic equation:
$17y^2 - 16y + 4 - 16 = 0$ $17y^2 - 16y - 12 = 0$

STEP 7

Solve the quadratic equation $17y^2 - 16y - 12 = 0$ using the quadratic formula:
$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 17$ , $b = -16$ , and $c = -12$ .

STEP 8

Calculate the discriminant and solve for $y$ :
$b^2 - 4ac = (-16)^2 - 4 \times 17 \times (-12)$ $= 256 + 816$ $= 1072$
$y = \frac{16 \pm \sqrt{1072}}{34}$

STEP 9

Calculate the square root and find the values of $y$ :
$\sqrt{1072} \approx 32.740$
$y_1 = \frac{16 + 32.740}{34} \approx 1.448$ $y_2 = \frac{16 - 32.740}{34} \approx -0.491$

STEP 10

Substitute $y_1$ and $y_2$ back into $x = 2 - 4y$ to find corresponding $x$ values:
For $y_1 = 1.448$ :
$x_1 = 2 - 4(1.448)$ $x_1 = 2 - 5.792$ $x_1 = -3.792$
For $y_2 = -0.491$ :
$x_2 = 2 - 4(-0.491)$ $x_2 = 2 + 1.964$ $x_2 = 3.964$

STEP 11

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

1. The system of equations consists of a circle equation x2+y2=16x^2 + y^2 = 16x2+y2=16 and a linear equation x+4y=2x + 4y = 2x+4y=2.2. The solution involves finding the intersection points of the circle and the line.3. The solutions should be rounded to three decimal places.

STEP 2

1. Solve the linear equation for one variable.2. Substitute the expression from the linear equation into the circle equation.3. Solve the resulting quadratic equation.4. Find the corresponding yyy values for each xxx.5. Round the solutions to three decimal places.

STEP 3

Solve the linear equation x+4y=2x + 4y = 2x+4y=2 for xxx:x=2−4y x = 2 - 4y x=2−4y

STEP 4

Substitute x=2−4yx = 2 - 4yx=2−4y into the circle equation x2+y2=16x^2 + y^2 = 16x2+y2=16:(2−4y)2+y2=16 (2 - 4y)^2 + y^2 = 16 (2−4y)2+y2=16

STEP 5

Expand and simplify the equation:(2−4y)2=4−16y+16y2 (2 - 4y)^2 = 4 - 16y + 16y^2 (2−4y)2=4−16y+16y2 4−16y+16y2+y2=16 4 - 16y + 16y^2 + y^2 = 16 4−16y+16y2+y2=16 17y2−16y+4=16 17y^2 - 16y + 4 = 16 17y2−16y+4=16

STEP 6

Move all terms to one side to form a quadratic equation:17y2−16y+4−16=0 17y^2 - 16y + 4 - 16 = 0 17y2−16y+4−16=0 17y2−16y−12=0 17y^2 - 16y - 12 = 0 17y2−16y−12=0

STEP 7

Solve the quadratic equation 17y2−16y−12=017y^2 - 16y - 12 = 017y2−16y−12=0 using the quadratic formula:y=−b±b2−4ac2a y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} y=2a−b±b2−4ac​​where a=17a = 17a=17, b=−16b = -16b=−16, and c=−12c = -12c=−12.

STEP 8

Calculate the discriminant and solve for yyy:b2−4ac=(−16)2−4×17×(−12) b^2 - 4ac = (-16)^2 - 4 \times 17 \times (-12) b2−4ac=(−16)2−4×17×(−12) =256+816 = 256 + 816 =256+816 =1072 = 1072 =1072y=16±107234 y = \frac{16 \pm \sqrt{1072}}{34} y=3416±1072​​

STEP 9

STEP 10

STEP 11

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The system of equations consists of a circle equation $x^2 + y^2 = 16$ and a linear equation $x + 4y = 2$ .
2. The solution involves finding the intersection points of the circle and the line.
3. The solutions should be rounded to three decimal places.

1. Solve the linear equation for one variable.
2. Substitute the expression from the linear equation into the circle equation.
3. Solve the resulting quadratic equation.
4. Find the corresponding $y$ values for each $x$ .
5. Round the solutions to three decimal places.

Solve the linear equation $x + 4y = 2$ for $x$ :
$x = 2 - 4y$

Substitute $x = 2 - 4y$ into the circle equation $x^2 + y^2 = 16$ :
$(2 - 4y)^2 + y^2 = 16$

Expand and simplify the equation:
$(2 - 4y)^2 = 4 - 16y + 16y^2$ $4 - 16y + 16y^2 + y^2 = 16$ $17y^2 - 16y + 4 = 16$

Move all terms to one side to form a quadratic equation:
$17y^2 - 16y + 4 - 16 = 0$ $17y^2 - 16y - 12 = 0$

Solve the quadratic equation $17y^2 - 16y - 12 = 0$ using the quadratic formula:
$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 17$ , $b = -16$ , and $c = -12$ .

Calculate the discriminant and solve for $y$ :
$b^2 - 4ac = (-16)^2 - 4 \times 17 \times (-12)$ $= 256 + 816$ $= 1072$
$y = \frac{16 \pm \sqrt{1072}}{34}$