Dashboard

Algebra

Solve · Slope · Slope-intercept
CORD — C Concept O Operation · Desmos R Recognition D Drill
L1 Solve linear equation (1 variable) — distribute, combine, isolate Drill 35
Desmos
  1. Move all terms to one side: f(x) = LHS - RHS.
  2. Plot y = f(x) → click the x-intercept. Desmos labels the solution.
  3. Absolute-value: type (1/2)|x - 2| - 3 = 1 as-is → Desmos draws vertical lines at the solutions (x = −6, 10). Read them off.
Lesson · Desmos for Algebra →
Concept
  • Order: 1) distribute, 2) combine like terms each side, 3) move variables to one side, 4) isolate.
  • Negative-sign trap: $-(x - 3) = -x + 3$, NOT $-x - 3$.
  • Coefficient is the rate (per-unit). Constant is the starting value / fixed fee.
  • Asked for $3x + 8$, not $x$? Solve for what's asked — don't isolate $x$ if the question never asks for it.
"$a$ such that no solution / inf solutions" — same classification logic as L9 systems (you'll see this Rule later).
RecognitionDrops the negative when distributing — writes -(x-3) as -x-3 instead of -x+3.
L2 Slope — find from 2 points / table / equation Drill 39
Desmos
  1. Click + → table. Enter the points: $x_1$ column = x-values, $y_1$ column = y-values.
  2. New line: type y_1 ~ a x_1 + b (the squiggle ~ is the key — it's a regression).
  3. Desmos prints a = slope and b = y-intercept below. Read off a.
  4. For a single equation: type it in Desmos and click x and y intercepts → slope = $\dfrac{\Delta y}{\Delta x}$ between them.
Lesson · Desmos for Algebra (Slope) →
Concept
  • 2 points $(x_1, y_1), (x_2, y_2)$: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$.
  • Table: $\dfrac{\Delta y}{\Delta x}$ between any two rows (slope is constant for linear).
  • Equation $y = mx + b$: slope is $m$. $ax + by = c$: slope is $-a/b$.
  • Horizontal line $y = k$ → slope $= 0$. Vertical line $x = k$ → slope undefined.
Why regression wins: no picking 2-of-3 points, no fraction errors, y-intercept free. Same ~ syntax later for quadratic + exponential.
RecognitionInverts run over rise, or subtracts the coordinates in opposite order between numerator and denominator.
L3 y = mx + b — slope-intercept · build, read, evaluate Drill 33
Desmos
  1. Build from points: use L2 regression — y_1 ~ a x_1 + b → reads both $m$ (=$a$) and the y-intercept $b$ at once.
  2. Verify a candidate equation: type it; type the given points as $(x_0, y_0)$; check the line passes through each.
  3. Match a graph in the question: type the candidate $y = m x + b$, compare to the figure visually.
Concept
  • $y = mx + b$: $m$ = slope, $b$ = $y$-intercept $(0, b)$.
  • Build from slope + y-int: plug both in directly.
  • Build from 2 points: compute $m$ first, then use one point to solve $b$.
  • Read from table: $b$ = $y$ at $x = 0$. $m$ = $\Delta y / \Delta x$.
  • From point-slope: $y - y_0 = m(x - x_0)$ → expand to $y = mx + b$.
RecognitionConfuses the y-intercept b with the slope m when reading y = mx + b.
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Algebra

Parallel / perp · Standard form · Linear function f(x)
L4 Parallel & perpendicular lines Drill 9
Desmos
  1. Algebraic first: compute the slope of each line. For $ax + by = c$, slope = $-a/b$.
  2. Solve unknown: if line $ax - 6y = 3$ is ⊥ to $3x + 2y = 7$, slopes are $a/6$ and $-3/2$. Set $\dfrac{a}{6} \cdot \dfrac{-3}{2} = -1$ → solve → $a = 4$.
  3. Or vector regression: type [(a/6)*(-3/2)] ~ [-1] → Desmos prints a = 4.
  4. Verify: plot both completed lines — check they cross at 90°.
Lesson · Perpendicular Lines →
Concept
  • Parallel: same slope. $m_1 = m_2$.
  • Perpendicular: opposite reciprocal. $m_1 \cdot m_2 = -1$, so $m_2 = -1/m_1$.
  • e.g. ⊥ to slope $2/3$ is slope $-3/2$.
  • $ax + by = c$ form: slope = $-a/b$. Useful when not given in $y = mx + b$.
  • Horizontal ⊥ vertical: $y = c$ ⊥ $x = k$. Slopes: $0$ and undefined.
Reciprocal AND sign flip. The negative of $2/3$ is just $-2/3$ — that's NOT the perpendicular slope. You must FLIP the fraction too.
RecognitionNegates the slope but forgets to flip the fraction, picking -2/3 instead of -3/2.
L5 Standard form ax + by = c — x- and y-intercepts Drill 15
Desmos
  1. Type the equation as-is, no rearranging: 3x + 4y = 24. Desmos draws the line.
  2. Click the line where it crosses the x-axis → Desmos labels the point $(x, 0)$.
  3. Click the line where it crosses the y-axis → Desmos labels the point $(0, y)$.
  4. From 3 points (not an equation): build a table with the points → run y_1 ~ a x_1 + b → click both intercepts on the regression line (lesson #3 in the book).
Lesson · Intercepts (3 points → regression) →
Concept
  • $x$-intercept: set $y = 0$ → $x = c/a$.
  • $y$-intercept: set $x = 0$ → $y = c/b$.
  • Slope from standard form: $m = -a/b$.
  • Convert to slope-int: $y = -\dfrac{a}{b}x + \dfrac{c}{b}$.
RecognitionSwaps the x- and y-intercepts, dividing c by the wrong coefficient.
L6 Linear function f(x) — evaluate, compose, find missing constants Drill 21
Desmos
  1. Define f(x) = m x + b in slot 1.
  2. Type f(5) on new line → Desmos prints the value.
  3. For "$f(a) = k$, find $a$": plot y = f(x) and y = k, click intersection.
  4. Two unknown constants: e.g. $f(x) = ax - b$, given $f(1) = 5$ and $ab = 6$ — use vector regression: [a - b, a*b] ~ [5, 6] → Desmos prints $a$ and $b$ as regression parameters.
Concept
  • Evaluate: plug $a$ into the rule.
  • Composition $f(g(x))$: plug $g(x)$ wherever $x$ appears.
  • Find missing constant like $f(x) = mx + a$, $f(4) = 31$: plug $4$, solve for $a$.
  • $f(0)$ always = y-intercept = constant term.
  • $f(cx) = x - k$ tricks: set the input = some number to solve for $f$ at that input.
RecognitionSolves for the input when the question asks for f(a).
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Algebra

Word setup · Interpret in context
L7 Word problem → linear equation setup Drill 87
Desmos
  1. After setup, type the equation. If 1 unknown: Desmos draws a vertical line at the solution — compare to answer choices or click for label.
  2. "How many years until total = X" → type $y = X$ and click intersection.
  3. For 2-variable setup, drag sliders to satisfy all conditions.
Concept
  • Fixed fee + per-unit rate: $y = (\text{rate}) \cdot x + (\text{fee})$. e.g. $36 + 19m$.
  • Arithmetic sequence ("first item costs A, each additional B"): $T(n) = A + (n - 1) \cdot B$. Expand to slope-int: $T(n) = (A - B) + Bn$.
  • "$a$ more than $b$": $a + b$. "$3$ times as many": $3x$.
  • "At a rate of $r$ per $t$": total $= rt$.
  • Mixture / weighted: $a \cdot x + b \cdot y$ = (rate)(quantity) sum.
Define variables FIRST. "Let $x$ = pounds of A" before writing the equation.
RecognitionWrites the equation before defining variables, then mixes up which quantity the rate multiplies.
L8 Interpret slope · intercept · coefficient in context Drill 18
Desmos
  1. Plot the equation. Read units off the axes.
  2. Slope visual: rise per unit run. Match units (e.g. dollars per hour).
  3. $y$-int visual: the value at $x = 0$ — usually a starting/baseline value.
Concept
  • Slope = rate of change. Units: $\dfrac{\text{output unit}}{\text{input unit}}$.
  • $y$-intercept = initial value (at $x = 0$). Fixed cost / starting amount / baseline.
  • $x$-intercept = when output is zero. e.g. when fuel runs out, balance hits 0.
  • Negative slope: decreasing rate. e.g. balance going down.
  • Constant term in $ax + b$: the part not depending on $x$ — usually a fee or baseline.
Always answer with units. "Dollars per hour" — not just "the rate".
RecognitionConfuses slope's rate with the intercept's starting value.
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Algebra

Systems · Inequalities
L9 System of linear equations — solve + # of solutions Drill 68
Desmos
  1. Vector regression: for $5x + 3y = 45$, $2x - 4y = -8$, type one line: [5x_1 + 3y_1, 2x_1 - 4y_1] ~ [45, -8]
  2. Desmos prints REGRESSION PARAMETERS: $x_1 = 6$, $y_1 = 5$, RMSE $= 0$ → that's your solution.
  3. Why this beats clicking intersection: works even when the intersection is off-screen, gives exact fractions/decimals, and doesn't need zoom-adjusting.
  4. Backup (standard method): type both equations as-is; click the intersection dot.
Lesson · Vector Regression →
Lesson · Click Intersection (standard) →
Concept
  • Substitute: solve one eq for $y$, plug into the other.
  • Eliminate: add/sub equations to cancel a variable. Multiply first to match coefficients.
  • Asked for $x + y$ or $x - y$ only: add/subtract the two equations directly — often skips solving each.
  • Classify by coefficient ratios for $ax + by = c$ and $a'x + b'y = c'$:
      • 1 solution: $a/a' \neq b/b'$
      • 0 solutions (parallel): $a/a' = b/b' \neq c/c'$
      • ∞ solutions (identical): $a/a' = b/b' = c/c'$
For "find $k$ such that the system has no solution / inf solutions" — set the slopes equal. Then check the constant ratio to distinguish parallel vs identical.
RecognitionMatches slopes but forgets constants separate no solution from infinite solutions.
L10 Linear inequality — solve · graph region · setup constraint Drill 46
Desmos
  1. Type the inequality directly: 3x + 4y <= 24 (use $\leq$ button).
  2. Desmos auto-shades the solution region.
  3. For systems: type all inequalities — overlap of shadings is the solution set.
  4. Test a point: drag it into the shaded region to verify.
Lesson · Inequalities →
Concept
  • Solve like an equation — but flip $\leq$ to $\geq$ when multiplying/dividing by negative.
  • "At least" = $\geq$. "At most" = $\leq$. "More than" = $>$. "Less than" = $<$.
  • "Between $a$ and $b$, inclusive": $a \leq x \leq b$.
  • "No more than $X$": total $\leq X$.
  • Region check: plug a test point into the inequality — if true, that side is shaded.
Whole number constraint: "Max packages" — round DOWN to integer. "Min packages" → round UP.
RecognitionForgets the sign flip after multiplying or dividing by a negative.
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