The image you provided contains a set of simultaneous equations from an extension exercise in algebra. These problems are designed to test skills in solving systems of linear and nonlinear equations. Below, I will explain the general approach to solving such equations and provide a detailed solution for one of the problems as an example.
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General Approach to Solving Simultaneous Equations
1.
Identify the Type of Equations:
-
Linear Equations: Equations where all variables have a degree of 1 (e.g., \( ax + by = c \)).
-
Nonlinear Equations: Equations involving higher powers of variables or other non-linear terms (e.g., \( x^2 + y = 5 \)).
2.
Choose a Method:
-
Substitution Method: Solve one equation for one variable and substitute it into the other equation.
-
Elimination Method: Add or subtract the equations to eliminate one variable.
-
Graphical Method: Plot the equations and find the intersection point(s).
-
Matrix Method: Use matrices and row operations to solve the system.
3.
Solve Step-by-Step:
- Simplify the equations if necessary.
- Apply the chosen method to find the values of the variables.
- Verify the solutions by substituting them back into the original equations.
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Example Problem: Solve Problem #1
#### Problem:
\[
\begin{cases}
x + y = 4 \\
3x - 2y = 7
\end{cases}
\]
#### Solution:
1.
Label the Equations:
\[
\text{(1)} \quad x + y = 4
\]
\[
\text{(2)} \quad 3x - 2y = 7
\]
2.
Use the Substitution Method:
- From Equation (1), solve for \( y \):
\[
y = 4 - x
\]
- Substitute \( y = 4 - x \) into Equation (2):
\[
3x - 2(4 - x) = 7
\]
3.
Simplify and Solve for \( x \):
\[
3x - 2(4 - x) = 7
\]
\[
3x - 8 + 2x = 7
\]
\[
5x - 8 = 7
\]
\[
5x = 15
\]
\[
x = 3
\]
4.
Find \( y \):
- Substitute \( x = 3 \) back into \( y = 4 - x \):
\[
y = 4 - 3 = 1
\]
5.
Verify the Solution:
- Check Equation (1):
\[
x + y = 3 + 1 = 4 \quad \text{(True)}
\]
- Check Equation (2):
\[
3x - 2y = 3(3) - 2(1) = 9 - 2 = 7 \quad \text{(True)}
\]
The solution is consistent with both equations.
#### Final Answer for Problem #1:
\[
\boxed{(x, y) = (3, 1)}
\]
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Additional Notes:
- For more complex problems (e.g., nonlinear equations or systems with fractions), the elimination or substitution methods may need to be adapted, or matrix methods might be more efficient.
- Always verify your solutions by substituting them back into the original equations to ensure accuracy.
If you need solutions for other specific problems from the list, feel free to ask!
Parent Tip: Review the logic above to help your child master the concept of simultaneous equations worksheet.
