Question

Solve the following system of equations for x to the nearest hundredth : y + 2x + 1 = 0; 4y - 4x ^ 2 - 12x = - 7

Answer 1

We can start by isolating y in the first equation:

y + 2x + 1 = 0

y = -2x - 1

Then, we can substitute this expression for y into the second equation:

4y - 4x^2 - 12x = -7

4(-2x - 1) - 4x^2 - 12x = -7

-8x - 4 - 4x^2 - 12x = -7

-4x^2 - 20x - 3 = 0

We can solve for x using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

Where a = -4, b = -20, and c = -3. Substituting these values, we get:

x = (-(-20) ± sqrt((-20)^2 - 4(-4)(-3))) / 2(-4)

x = (20 ± sqrt(400 - 48)) / (-8)

x = (20 ± sqrt(352)) / (-8)

x = (20 ± 18.78) / (-8)

So the solutions are:

x ≈ -0.11 or x ≈ -4.39

Therefore, the system of equations has two solutions for x: x ≈ -0.11 and x ≈ -4.39 (rounded to the nearest hundredth).

Answer 2

We can start by solving the first equation for y:

• y + 2x + 1 = 0
• y = -2x - 1

We can then substitute this expression for y into the second equation:

• 4y - 4x^2 - 12x = -7
• 4(-2x - 1) - 4x^2 - 12x = -7
• -8x - 4 - 4x^2 - 12x = -7
• -4x^2 - 20x + 3 = 0

We can solve for x using the quadratic formula:

• x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = -4, b = -20, and c = 3.

• x = (-(-20) ± sqrt((-20)^2 - 4(-4)(3))) / 2(-4)
• x = (20 ± sqrt(400 + 48)) / (-8)
• x = (20 ± sqrt(448)) / (-8)
• x = (20 ± 4sqrt(7)) / (-8)
• x ≈ -0.85 or x ≈ -2.93

Verification:

To verify these solutions, we can substitute them back into the original equations:

For x ≈ -0.85:

• y + 2x + 1 = 0
• y + 2(-0.85) + 1 = 0
• y ≈ 0.7
• 4y - 4x^2 - 12x = -7
• 4(0.7) - 4(-0.85)^2 - 12(-0.85) = -7
• -2.8 - 2.89 + 10.2 = -7

This checks out, so x ≈ -0.85 is a valid solution.

• For x ≈ -2.93:
• y + 2x + 1 = 0
• y + 2(-2.93) + 1 = 0
• y ≈ 4.86
• 4y - 4x^2 - 12x = -7
• 4(4.86) - 4(-2.93)^2 - 12(-2.93) = -7
• 19.44 - 33.45 + 35.16 = -7.85

This does not check out, so x ≈ -2.93 is not a valid solution.

Therefore, the only solution to the system of equations to the nearest hundredth is x ≈ -0.85, and the corresponding value of y is y ≈ 0.7.