Today's soundtrack is dc Talk: Supernatural, the album that got me into the band in the first place. I remember being blown away by the variety of styles throughout the album, and by the raw vocal talent of the three singers. There are so many great songs on this album; it's hard-hitting at times, contemplative at others, and always good.
I started the morning by releasing "The Colours of Space" on Bandcamp and YouTube - gotta start the year off right!
In the last math lesson, I learned how to solve systems of equations graphically. Today, I'm learning how to solve systems of equations algebraically. I'll be learning how to solve both "linear-quadratic systems and quadratic-quadratic systems" (Pearson's Pre-Calculus 11, p. 391).
As I learned in the last lesson, a linear-quadratic equation "may have 0, 1, or 2 solutions" (p. 392), and a quadratic-quadratic equation may have 0, 1, 2, or infinite solutions. Each solution is simply a coordinate (or "ordered pair") where lines overlap.
Solving a Linear-Quadratic Equation
1. Set up the equations
a. Ensure the linear equation is in the slope-intercept form (y = mx + b), then isolate x
b. Ensure the quadratic equation is in general form (y = ax² + bx + c), then isolate y
​2. Use substitution to put the linear equation into the value of y in the quadratic equation
3. Solve for y either by factoring, completing the square, or using the quadratic formula (I prefer the latter)
4. Substitute the value[s] of y back into the linear equation that we got after isolating x in step 1a
5. Solve for x
​​The solution[s] will be in the format of 0, 1, or 2 ordered pair[s], using the value[s] of x that we got in step four as our x-value[s], and their respective y-value[s] as the y-value[s] of the ordered pair[s]. We can verify our solution[s] by using a graphing calculator.
Solving a Quadratic-Quadratic Equation
1. Set up the equations
a. Ensure both equations are in general form (y = ax² + bx + c)
b. Identify one equation as Equation 1, and identify the other as Equation 2
2. Use substitution to put the first equation into the value of y in the second equation
3. Simplify
4. Interpret the discriminant to determine whether the equation can be factored
5. If it can be factored, solve for x either by factoring, completing the square, or using the quadratic formula
6. Substitute the value[s] of x into Equation 1
7. Solve the first equation for y
​The solution[s] will be in the format of 0, 1 2, or infinite, ordered pair[s], using the value[s] of x that we got in step five as our x-value[s], and their respective y-value[s] as the y-value[s] of the ordered pair[s]. We can verify our solution[s] by using a graphing calculator.
Welp, that's enough math for today. Tomorrow, I'll take a crack at the assignment portion of this lesson, and I'll try to get some examples up.