To solve the given linear-quadratic system using the substitution method, we have the equations:

Question

To solve the given linear-quadratic system using the substitution method, we have the equations:
1. $ y = x^2 - 4x - 41 $ (quadratic equation)
2. $ y = 4 $ (linear equation)

We will substitute the value of $ y $ from the second equation into the first equation.

### Step 1: Substitute $ y = 4 $ into the quadratic equation

Substituting $ y = 4 $ into $ y = x^2 - 4x - 41 $:

$$
4 = x^2 - 4x - 41
$$

### Step 2: Rearrange the equation

Rearranging the equation gives:

$$
x^2 - 4x - 41 - 4 = 0
$$
$$
x^2 - 4x - 45 = 0
$$

### Step 3: Factor or use the quadratic formula

We need to solve the quadratic equation $ x^2 - 4x - 45 = 0 $. We can factor it:

$$
(x - 9)(x + 5) = 0
$$

### Step 4: Set each factor to zero

Setting each factor equal to zero gives us:

$$
x - 9 = 0 \quad \Rightarrow \quad x = 9
$$

$$
x + 5 = 0 \quad \Rightarrow \quad x = -5
$$

### Step 5: Find corresponding $ y $ values

Now we substitute $ x = 9 $ and $ x = -5 $ back into the linear equation $ y = 4 $:

- For $ x = 9 $:

$$
y = 4
$$

- For $ x = -5 $:

$$
y = 4
$$

### Step 6: Write the solutions

Thus, the solutions to the linear-quadratic system are:

1. $ (9, 4) $
2. $ (-5, 4) $

These are the points where the quadratic graph intersects the line $ y = 4 $.

make this like a 6th grader wrote it

GPT-4o mini · Answer

Sure! Let's simplify the explanation so a 6th grader can understand it.

We're trying to solve two equations:

$y = x^{2} - 4 x - 41$ (this one is a bit curvy called a quadratic)
$y = 4$ (this one is a straight line)

Step 1: Plug in $y = 4$

We know $y$ is 4 from the second equation. So, we put 4 in for $y$ in the first equation:

$4 = x^{2} - 4 x - 41$

Step 2: Move things around

Now, we want to make it a bit easier to solve. We’ll move 4 over to the other side:

$x^{2} - 4 x - 41 - 4 = 0$

So this becomes:

$x^{2} - 4 x - 45 = 0$

Step 3: Solve the equation

Next, we need to solve the equation $x^{2} - 4 x - 45 = 0$ . We can make this easier by breaking it down into two parts. We want two numbers that multiply to -45 and add up to -4. Those numbers are:

9 and -5

So, we can write it like this:

$(x - 9) (x + 5) = 0$

Step 4: Find the answers for $x$

Now we set each part to zero:

$x - 9 = 0$ means $x = 9$
$x + 5 = 0$ means $x = - 5$

Step 5: Get $y$

Both times we found $x$ , we can find $y$ using the second equation $y = 4$ :

When $x = 9$ , $y = 4$
When $x = - 5$ , $y = 4$

Step 6: Write the final answer

So the solutions to our equations, which are the points where the curvy line meets the straight line, are:

$(9, 4)$
$(- 5, 4)$

And that's it! Now we know where the two equations meet!