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Solving systems of linear and nonlinear equations using multiple methods
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A system of equations is a set of two or more equations with the same variables. A solution is a set of values that satisfies all equations simultaneously.
Linear System (2 equations, 2 unknowns):
ax + by = c \\ dx + ey = f \end{cases}$$ **Nonlinear System** (may include quadratics, circles, etc.): $$\begin{cases} y = x^2 \\ y = 2x + 3 \end{cases}$$ ## Number of Solutions A system can have: - **One solution**: Lines/curves intersect at exactly one point - **No solution**: Lines/curves are parallel (never intersect) - **Infinitely many solutions**: Lines/curves coincide (same graph) ## Method 1: Substitution **Steps:** 1. Solve one equation for one variable 2. Substitute into the other equation 3. Solve for the remaining variable 4. Back-substitute to find the other variable **Best for:** When one equation is already solved for a variable, or easily can be ### Example $$\begin{cases} y = 2x - 1 \\ 3x + y = 9 \end{cases}$$ Substitute $y = 2x - 1$ into second equation: $$3x + (2x - 1) = 9$$ $$5x - 1 = 9$$ $$5x = 10$$ $$x = 2$$ Then: $y = 2(2) - 1 = 3$ **Solution:** $(2, 3)$ ## Method 2: Elimination (Addition) **Steps:** 1. Multiply equations (if needed) to make coefficients of one variable opposites 2. Add equations to eliminate that variable 3. Solve for the remaining variable 4. Substitute back to find the other variable **Best for:** When coefficients are easy to manipulate ### Example $$\begin{cases} 2x + 3y = 8 \\ x - 3y = 1 \end{cases}$$ Add the equations (the $3y$ terms cancel): $$3x = 9$$ $$x = 3$$ Substitute into second equation: $$3 - 3y = 1$$ $$-3y = -2$$ $$y = \frac{2}{3}$$ **Solution:** $(3, \frac{2}{3})$ ## Method 3: Graphing **Steps:** 1. Graph both equations 2. Find intersection point(s) 3. Verify by substitution **Best for:** Visual understanding, approximate solutions ## Systems with 3 Variables For three variables $(x, y, z)$, you need three equations: $$\begin{cases} ax + by + cz = d \\ ex + fy + gz = h \\ ix + jy + kz = l \end{cases}$$ **Strategy:** Use elimination to reduce to 2 variables, then solve ## Nonlinear Systems For systems involving quadratics, circles, or other curves: - Substitution is usually best - May have 0, 1, 2, or more solutions - Graph to visualize ### Example: Line and Parabola $$\begin{cases} y = x^2 - 4 \\ y = 2x - 1 \end{cases}$$ Substitute: $x^2 - 4 = 2x - 1$ $$x^2 - 2x - 3 = 0$$ $$(x - 3)(x + 1) = 0$$ $$x = 3 \text{ or } x = -1$$ Solutions: $(3, 5)$ and $(-1, -3)$ ## Applications - **Mixture problems**: Combining solutions with different concentrations - **Distance/rate/time**: Two objects moving - **Business**: Supply and demand, break-even analysis - **Geometry**: Finding intersection pointsSolve the system using substitution:
Avoid these 4 frequent errors
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A stone is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a rate of cm/s. How fast is the area of the circle increasing when the radius is cm?
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Solve using substitution:
Step 1: Solve the first equation for :
Step 2: Substitute into the second equation:
Step 3: Find :
Step 4: Verify in both equations: โ โ
Answer:
Solve using elimination:
Solve using elimination:
Step 1: Make coefficients of one variable opposites. Multiply first equation by 2 and second by 5:
4x + 10y = 26 \\ 15x - 10y = -20 \end{cases}$$ Step 2: Add to eliminate $y$: $$19x = 6$$ $$x = \frac{6}{19}$$ Step 3: Substitute into first original equation: $$2\left(\frac{6}{19}\right) + 5y = 13$$ $$\frac{12}{19} + 5y = 13$$ $$5y = 13 - \frac{12}{19} = \frac{247 - 12}{19} = \frac{235}{19}$$ $$y = \frac{235}{95} = \frac{47}{19}$$ Step 4: Verify: $$2\left(\frac{6}{19}\right) + 5\left(\frac{47}{19}\right) = \frac{12 + 235}{19} = \frac{247}{19} = 13$$ โ **Answer:** $\left(\frac{6}{19}, \frac{47}{19}\right)$Solve the nonlinear system:
Solve the system (circle and line):
Step 1: Substitute into the first equation:
Solve the system: 2x + 3y = 12 and x - y = 1
Step 1: Use substitution method - solve second equation for x: x - y = 1 x = y + 1
Step 2: Substitute into first equation: 2(y + 1) + 3y = 12 2y + 2 + 3y = 12 5y + 2 = 12 5y = 10 y = 2
Step 3: Find x using x = y + 1: x = 2 + 1 = 3
Step 4: Verify in both equations: 2(3) + 3(2) = 6 + 6 = 12 โ 3 - 2 = 1 โ
Answer: x = 3, y = 2 or (3, 2)
Solve the system: xยฒ + yยฒ = 25 and y = x + 1
Step 1: This is a nonlinear system (circle and line) Circle: xยฒ + yยฒ = 25 (radius 5, centered at origin) Line: y = x + 1
Step 2: Substitute y = x + 1 into the circle equation: xยฒ + (x + 1)ยฒ = 25
Step 3: Expand and simplify: xยฒ + xยฒ + 2x + 1 = 25 2xยฒ + 2x + 1 = 25 2xยฒ + 2x - 24 = 0 xยฒ + x - 12 = 0
Step 4: Factor: (x + 4)(x - 3) = 0 x = -4 or x = 3
Step 5: Find corresponding y values: If x = -4: y = -4 + 1 = -3 If x = 3: y = 3 + 1 = 4
Step 6: Verify both solutions: (-4, -3): (-4)ยฒ + (-3)ยฒ = 16 + 9 = 25 โ (3, 4): 3ยฒ + 4ยฒ = 9 + 16 = 25 โ
Answer: (-4, -3) and (3, 4)
Step 2: Factor:
Step 3: Find corresponding values:
Step 4: Verify both solutions:
For : โ โ
For : โ โ
Answer: Two solutions: and
Interpretation: The line intersects the circle at two points.