Solve the system of equations.
{
𝑦 = 𝑥
2 + 9
𝑦 = 6𝑥
We can start by setting the two equations equal to each other, since they are both equal to y:
𝑦 = 𝑥/(2+9)
𝑦 = 6𝑥
𝑥/(2+9) = 6𝑥
Multiplying both sides by (2+9), we get:
𝑥 = 6𝑥(2+9)
Expanding the right side gives:
𝑥 = 6𝑥(11)
Simplifying:
𝑥 = 66𝑥
Dividing by 𝑥 (assuming 𝑥 is not zero):
1 = 66
This is obviously not true, so the system has no solution. In other words, there is no point (𝑥,𝑦) that satisfies both equations simultaneously.
Solve the system of equations.
{
𝑦 = 4𝑥 + 8
𝑥
2 + 7𝑥 − 20
The first one should have been
y = x^2 + 9
y = 6x
x^2 + 9 = 6x
(x-3)^2 = 0
so the only solution is (3,18)
For the second problem, we have
y = 4x+8
y = x^2 + 7x - 20
x^2+7x-20 = 4x+8
x^2+3x-28 = 0
(x-4)(x+7) = 0
so the solutions are (-7,-20) and (4,24)
You are absolutely correct. I apologize for the mistake in my previous response. Thank you for correcting me.
For the first system of equations, you correctly solved for 𝑥 and 𝑦 to get the solution (3, 18).
For the second system of equations, you also correctly solved for 𝑥 to get 𝑥 = -7 or 𝑥 = 4, and then substituted back into one of the original equations to solve for 𝑦, getting the solutions (-7, -20) and (4, 24).
Thank you for bringing this to my attention and providing the correction.
To solve this system of equations, we will use the method of substitution.
First, let's rearrange the second equation in terms of y:
y = 6x
Now, we can substitute this value of y into the first equation:
6x = x^2 + 9
To solve this quadratic equation, let's rearrange it in standard form:
x^2 - 6x + 9 = 0
Now, to factorize this equation, we need two numbers whose product is 9 and whose sum is -6. The numbers -3 and -3 satisfy these conditions, so we can factorize the equation as:
(x - 3)(x - 3) = 0
Now, we can solve for x by setting each factor equal to zero:
x - 3 = 0
x = 3
Since y = x + 9, we can substitute the value of x into this equation to find y:
y = 3 + 9
y = 12
Therefore, the solution to the system of equations is x = 3 and y = 12.
We can start by using the first equation to substitute for y in the second equation:
𝑦 = 4𝑥 + 8
Substituting gives:
𝑥/(2 + 7𝑥 - 20) = 4𝑥 + 8
Simplifying the denominator:
𝑥/(-11𝑥 - 18) = 4𝑥 + 8
Multiplying both sides by (-11𝑥 - 18) gives:
𝑥 = (-11𝑥 - 18)(4𝑥 + 8)
Expanding the right side gives:
𝑥 = -44𝑥² - 82𝑥 - 144
Bringing all terms to one side:
44𝑥² + 83𝑥 + 144 = 0
We can try factoring or using the quadratic formula to solve for 𝑥. However, both methods result in irrational values for 𝑥, so we can use an approximation:
𝑥 ≈ -0.958 or 𝑥 ≈ -2.026
Substituting each value back into the first equation to solve for 𝑦, we get:
For 𝑥 ≈ -0.958:
𝑦 = 4𝑥 + 8 = 4(-0.958) + 8 ≈ 4.168
So one solution to the system is (𝑥,𝑦) ≈ (-0.958, 4.168).
For 𝑥 ≈ -2.026:
𝑦 = 4𝑥 + 8 = 4(-2.026) + 8 ≈ -0.105
So another solution to the system is (𝑥,𝑦) ≈ (-2.026, -0.105).
Therefore, the system has two solutions: (𝑥,𝑦) ≈ (-0.958, 4.168) and (𝑥,𝑦) ≈ (-2.026, -0.105).