To solve the system of equations using the substitution method:

System of equations:

1. $x^2 + y = 4$
2. $x - y + 2 = 0$

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Step 1: Solve one equation for one variable

From the second equation: $x - y + 2 = 0$ $y = x + 2$

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Step 2: Substitute $y = x + 2$ into the first equation

Substitute into $x^2 + y = 4$: $x^2 + (x + 2) = 4$ Simplify: $x^2 + x + 2 = 4$ $x^2 + x - 2 = 0$

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Step 3: Solve the quadratic equation $x^2 + x - 2 = 0$

Factorize: $x^2 + x - 2 = (x + 2)(x - 1) = 0$ Thus: $x = -2 \quad \text{or} \quad x = 1$

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Step 4: Find corresponding $y$-values

Using $y = x + 2$:

• For $x = -2$: $y = -2 + 2 = 0$ Solution: $(-2, 0)$

• For $x = 1$: $y = 1 + 2 = 3$ Solution: $(1, 3)$

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Final Answer:

• Smaller $x$-value: $(x, y) = (-2, 0)$
• Larger $x$-value: $(x, y) = (1, 3)$

Would you like further details or clarification on any step?

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Related Questions:

1. How do you check solutions in a system of equations?
2. What are other methods for solving systems of equations, such as elimination?
3. How do you solve non-linear systems of equations graphically?
4. What are the advantages of substitution over other methods in certain cases?
5. How can quadratic equations like $x^2 + x - 2 = 0$ be solved using the quadratic formula?

Tip:

When solving systems, always verify your solutions by substituting them back into both original equations!