Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To solve the given system of equations for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ 3^{2y + x} = 3^5 \][/tex]
[tex]\[ y = 2x(x - 3) - 8 \][/tex]
Let's proceed step by step:
### Step 1: Solve the first equation
We have the equation:
[tex]\[ 3^{2y + x} = 3^5 \][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[ 2y + x = 5 \][/tex]
This simplifies to:
[tex]\[ 2y = 5 - x \][/tex]
[tex]\[ y = \frac{5 - x}{2} \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] in the second equation
We now substitute [tex]\( y = \frac{5 - x}{2} \)[/tex] into the second equation:
[tex]\[ y = 2x(x - 3) - 8 \][/tex]
Substituting [tex]\( y \)[/tex]:
[tex]\[ \frac{5 - x}{2} = 2x(x - 3) - 8 \][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]
First, clear the fraction by multiplying both sides by 2:
[tex]\[ 5 - x = 4x(x - 3) - 16 \][/tex]
Simplify the right side:
[tex]\[ 5 - x = 4x^2 - 12x - 16 \][/tex]
Rearrange all terms to one side to form a quadratic equation:
[tex]\[ 4x^2 - 12x - 16 + x - 5 = 0 \][/tex]
[tex]\[ 4x^2 - 11x - 21 = 0 \][/tex]
### Step 4: Solve the quadratic equation
To solve [tex]\( 4x^2 - 11x - 21 = 0 \)[/tex], we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 4 \)[/tex], [tex]\( b = -11 \)[/tex], and [tex]\( c = -21 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-11)^2 - 4(4)(-21) = 121 + 336 = 457 \][/tex]
Thus:
[tex]\[ x = \frac{11 \pm \sqrt{457}}{8} \][/tex]
### Step 5: Calculate the values
The solutions for [tex]\( x \)[/tex] are:
[tex]\[ x_1 = \frac{11 + \sqrt{457}}{8} \][/tex]
[tex]\[ x_2 = \frac{11 - \sqrt{457}}{8} \][/tex]
### Step 6: Find corresponding [tex]\( y \)[/tex] values
Using [tex]\( y = \frac{5 - x}{2} \)[/tex], we find [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]:
For [tex]\( x_1 = \frac{11 + \sqrt{457}}{8} \)[/tex]:
[tex]\[ y_1 = \frac{5 - \frac{11 + \sqrt{457}}{8}}{2} = \frac{40 - 11 - \sqrt{457}}{16} = \frac{29 - \sqrt{457}}{16} \][/tex]
For [tex]\( x_2 = \frac{11 - \sqrt{457}}{8} \)[/tex]:
[tex]\[ y_2 = \frac{5 - \frac{11 - \sqrt{457}}{8}}{2} = \frac{40 - 11 + \sqrt{457}}{16} = \frac{29 + \sqrt{457}}{16} \][/tex]
### Final Solution
Therefore, the solutions for [tex]\( (x, y) \)[/tex] are:
[tex]\[ \left( \frac{11 + \sqrt{457}}{8}, \frac{29 - \sqrt{457}}{16} \right) \][/tex]
and
[tex]\[ \left( \frac{11 - \sqrt{457}}{8}, \frac{29 + \sqrt{457}}{16} \right) \][/tex]
[tex]\[ 3^{2y + x} = 3^5 \][/tex]
[tex]\[ y = 2x(x - 3) - 8 \][/tex]
Let's proceed step by step:
### Step 1: Solve the first equation
We have the equation:
[tex]\[ 3^{2y + x} = 3^5 \][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[ 2y + x = 5 \][/tex]
This simplifies to:
[tex]\[ 2y = 5 - x \][/tex]
[tex]\[ y = \frac{5 - x}{2} \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] in the second equation
We now substitute [tex]\( y = \frac{5 - x}{2} \)[/tex] into the second equation:
[tex]\[ y = 2x(x - 3) - 8 \][/tex]
Substituting [tex]\( y \)[/tex]:
[tex]\[ \frac{5 - x}{2} = 2x(x - 3) - 8 \][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]
First, clear the fraction by multiplying both sides by 2:
[tex]\[ 5 - x = 4x(x - 3) - 16 \][/tex]
Simplify the right side:
[tex]\[ 5 - x = 4x^2 - 12x - 16 \][/tex]
Rearrange all terms to one side to form a quadratic equation:
[tex]\[ 4x^2 - 12x - 16 + x - 5 = 0 \][/tex]
[tex]\[ 4x^2 - 11x - 21 = 0 \][/tex]
### Step 4: Solve the quadratic equation
To solve [tex]\( 4x^2 - 11x - 21 = 0 \)[/tex], we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 4 \)[/tex], [tex]\( b = -11 \)[/tex], and [tex]\( c = -21 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-11)^2 - 4(4)(-21) = 121 + 336 = 457 \][/tex]
Thus:
[tex]\[ x = \frac{11 \pm \sqrt{457}}{8} \][/tex]
### Step 5: Calculate the values
The solutions for [tex]\( x \)[/tex] are:
[tex]\[ x_1 = \frac{11 + \sqrt{457}}{8} \][/tex]
[tex]\[ x_2 = \frac{11 - \sqrt{457}}{8} \][/tex]
### Step 6: Find corresponding [tex]\( y \)[/tex] values
Using [tex]\( y = \frac{5 - x}{2} \)[/tex], we find [tex]\( y \)[/tex] for each [tex]\( x \)[/tex]:
For [tex]\( x_1 = \frac{11 + \sqrt{457}}{8} \)[/tex]:
[tex]\[ y_1 = \frac{5 - \frac{11 + \sqrt{457}}{8}}{2} = \frac{40 - 11 - \sqrt{457}}{16} = \frac{29 - \sqrt{457}}{16} \][/tex]
For [tex]\( x_2 = \frac{11 - \sqrt{457}}{8} \)[/tex]:
[tex]\[ y_2 = \frac{5 - \frac{11 - \sqrt{457}}{8}}{2} = \frac{40 - 11 + \sqrt{457}}{16} = \frac{29 + \sqrt{457}}{16} \][/tex]
### Final Solution
Therefore, the solutions for [tex]\( (x, y) \)[/tex] are:
[tex]\[ \left( \frac{11 + \sqrt{457}}{8}, \frac{29 - \sqrt{457}}{16} \right) \][/tex]
and
[tex]\[ \left( \frac{11 - \sqrt{457}}{8}, \frac{29 + \sqrt{457}}{16} \right) \][/tex]
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
