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Solving systems of linear and nonlinear equations using multiple methods
Learn step-by-step with practice exercises built right in.
A system of equations is a set of two or more equations with the same variables. A is a set of values that satisfies all equations simultaneously.
Linear System (2 equations, 2 unknowns):
Solve the system using substitution:
Solve using substitution:
Step 1: Solve the first equation for :
Solve using elimination:
Solve the nonlinear system:
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)
Avoid these 4 frequent errors
See how this math is used in the real world
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?
Step 2: Substitute into the second equation:
Step 3: Find :
Step 4: Verify in both equations: โ โ
Answer:
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 system (circle and line):
Step 1: Substitute into the first equation:
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.