To find the point of intersection of two simultaneous equations, you need to solve the system of equations. This can be done using various methods such as substitution, elimination, or graphical methods. Here's a step-by-step guide to each method:

1. Substitution Method:

• Solve one of the equations for one variable in terms of the other.
• Substitute this expression into the second equation.
• Solve for the remaining variable.
• Use the value of this variable to find the other variable by substituting back into one of the original equations.

Example:

$y = 2x + 1 \quad (1)$ $y = -x + 4 \quad (2)$

• From equation (1), substitute $y = 2x + 1$ into equation (2): $2x + 1 = -x + 4$
• Solve for $x$: $3x = 3 \quad \Rightarrow \quad x = 1$
• Substitute $x = 1$ into equation (1): $y = 2(1) + 1 = 3$
• So, the point of intersection is $(1, 3)$.

2. Elimination Method:

• Multiply or manipulate the equations to eliminate one of the variables by adding or subtracting the equations.
• Solve the resulting equation for the remaining variable.
• Use this value to find the other variable.

Example:

$2x + y = 5 \quad (1)$ $3x - y = 4 \quad (2)$

• Add the two equations to eliminate $y$: $(2x + y) + (3x - y) = 5 + 4$ $5x = 9 \quad \Rightarrow \quad x = \frac{9}{5}$
• Substitute $x = \frac{9}{5}$ into one of the original equations to solve for $y$: $2\left(\frac{9}{5}\right) + y = 5$ $\frac{18}{5} + y = 5 \quad \Rightarrow \quad y = 5 - \frac{18}{5} = \frac{7}{5}$
• The point of intersection is $\left(\frac{9}{5}, \frac{7}{5}\right)$.

3. Graphical Method:

• Plot both equations on a graph.
• The point where the two lines intersect is the solution to the system of equations.
• This method is useful for visualizing the relationship between the two equations but may be less accurate unless done carefully.

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Would you like more details on any of these methods?

Here are 5 related questions to expand the topic:

1. How do you solve simultaneous equations with more than two variables?
2. Can simultaneous equations have no solution or infinitely many solutions?
3. How do you solve non-linear simultaneous equations, like one quadratic and one linear?
4. What is the role of matrices in solving simultaneous equations?
5. How can you use software tools like graphing calculators or computer programs to solve systems of equations?

Tip:

When solving systems of equations, always check if the system is consistent (has a solution) or inconsistent (no solution), which can save time analyzing the situation.