Notes on Linear Equations and Systems

Slopes and Perpendicular Lines

• When two lines are perpendicular, their slopes are negative reciprocals of each other.

  • If one slope is $m<em>1$, the other slope is $m</em>2 = -\frac{1}{m_1}$.

  • This is the rule of perpendicular slopes: the product is $m<em>1 \cdot m</em>2 = -1$.

• Example context from the talk:

  • If a line has slope $m = -6$, a perpendicular line would have slope $m_2 = -\frac{1}{-6} = \frac{1}{6}$.

  • If a line slope is (discussed as the negative reciprocal relationship), you can determine the perpendicular slope using the same rule.

• Practical takeaway: knowing one slope lets you construct a line perpendicular to it by using the negative reciprocal.

Point-Slope Form and Converting to Slope-Intercept Form

• Point-slope form (uses a known point and the slope):

  • $y - y<em>1 = m\,(x - x</em>1)$

  • This form emphasizes a point on the line and the slope.

• Why this form is useful: it’s convenient when you know a point on the line and the slope, rather than the y-intercept.

• Converting to slope-intercept form (to graph easily or compare with y = mx + b):

  • Solve for y:

  • Starting from $y - y<em>1 = m\,(x - x</em>1)$

  • Expand to get $y = m\,x + (y<em>1 - m\,x</em>1)$

  • So the y-intercept is $b = y<em>1 - m\,x</em>1$.

• Quick example (illustrative):

  • Let slope $m = 3$ and a known point \((x1, y1) = (2, -1)\).

  • Point-slope form: $y + 1 = 3\,(x - 2)$

  • Convert: $y + 1 = 3x - 6 \Rightarrow y = 3x - 7$

  • So the slope-intercept form is $y = 3x - 7$ with intercept $b = -7$.

• Note on the lecture’s occasional wording: the discussion referenced a point-slope form and converting to a form that makes substitution and graphing more straightforward; the essential idea is shown above.

Systems of Linear Equations: What Changes When We Talk About Systems

• A system is two or more linear equations considered together.

• A solution to a system is a pair of values ((x, y)) that makes every equation in the system true.

• Graphically, each equation represents a line; the solution to the system is the point where the lines intersect.

• Notation and intuition:

  • A solution is an ordered pair, often written as ((x{}, y{})).

  • If you plug the pair into each equation, every equation should hold true (e.g., each left-hand side equals the right-hand side).

• Real-life analogy (donut shop example):

  • Let price of glazed donuts be $g$ and stuffed donuts be $s$.

  • One purchase gives an equation like $a\,g + b\,s =$(total), for some integers (a,b).

  • With one equation and two unknowns, you only get a relationship between the prices, not a single pair.

  • A second purchase (second equation) typically pins down a unique pair ((g, s)) if the equations are independent.

• Important special cases:

  • If two lines intersect, there is a unique solution (the intersection point).

  • If two lines are parallel (same slope, different intercept), there is no solution.

  • If two lines coincide (the same line, one is a scalar multiple of the other), there are infinitely many solutions.

• Visual takeaway: the solution set to a system corresponds to the intersection(s) of the lines represented by the equations.

Graphical Perspective and How to Solve by Graphing

• Graph each equation as a line; the intersection point is the solution.

• Practical workflow:

  • Pick two points on each line, draw the lines, and read the intersection.

  • Tools: Desmos, a graphing calculator, or grid notebooks.

  • The lecture demonstrated that Desmos can plot any equation form, not just slope-intercept form.

• Important caveats:

  • If the two lines do not intersect, there is no solution.

  • If the two lines lie on top of each other, there are infinitely many solutions.

Substitution Method for Solving Systems

• Concept: use one equation to express one variable in terms of the other, then substitute into the second equation.

• Step-by-step procedure:
  1) Choose one equation; solve for one variable (either x or y).
  2) Substitute that expression into the other equation.
  3) Solve for the remaining variable.
  4) Substitute back into either original equation to find the other variable.

• Worked example (consistent system to illustrate the method):

  • Consider the system
    $\begin{cases} 2x + y = 1 \ 4x + 3y = 3 \end{cases}$

  • Step 1: Solve the first equation for y:
    $y = 1 - 2x$

  • Step 2: Substitute into the second equation:
    $4x + 3(1 - 2x) = 3$

  • Step 3: Solve for x:
    $4x + 3 - 6x = 3 \Rightarrow -2x = 0 \Rightarrow x = 0$

  • Step 4: Solve for y using y = 1 - 2x:
    $y = 1 - 2(0) = 1$

  • Solution: ((x, y) = (0, 1)).

  • Verification: Substitute back into both equations to ensure they hold.

• The lecture’s stepwise narrative illustrated the same substitution idea, ending with a specific pair after solving for one variable and back-substituting for the other.

Quick Reference: Key Formulas and Concepts

• Perpendicular slopes: $m<em>1 \cdot m</em>2 = -1 \quad\text{or}\quad m<em>2 = -\frac{1}{m</em>1}$

• Point-slope to slope-intercept conversion: from $y - y<em>1 = m\,(x - x</em>1)$ to $y = m x + b$ with $b = y<em>1 - m x</em>1$

• System of equations representation: a set of linear equations whose common solution, when it exists, is the intersection of the corresponding lines.

• Solutions and cases:

  • Unique solution: one intersection point

  • No solution: parallel lines

  • Infinitely many solutions: coincident lines

• Substitution workflow recap:

  • Solve one equation for one variable; substitute into the other; solve; back-substitute.

Practical Tips and Study Notes

• Use graphing tools to visually confirm the solution; Desmos is a popular choice.

• When solving by substitution, keep track of signs and arithmetic carefully; small errors can lead to incorrect conclusions about the number of solutions.

• In real-life modeling with donuts or other items, set up equations carefully with counts as coefficients and totals as constants to capture the constraints.

• If a system looks underdetermined (fewer independent equations than unknowns), you may have infinitely many solutions; check whether the second equation is simply a multiple of the first.

• If a system looks inconsistent (no intersection), verify the equations for possible errors in data or modeling assumptions.

Summary of Core Ideas

• Linear equations describe lines; systems of linear equations describe multiple lines together.

• The solution to a system is the set of values that satisfy every equation in the system; graphically, the intersection of the lines.

• There are three main outcomes for two-equation systems: a unique solution (intersection point), no solution (parallel lines), or infinitely many solutions (coincident lines).

• The substitution method provides a concrete algebraic path to the solution by elimination of one variable, followed by back-substitution to obtain the remaining variable.

• Graphing offers a visual check and a practical way to approximate the solution; accuracy can be enhanced with algebraic methods.