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# Advanced Mathematics Support Programme ## Simultaneous Equations ### What are Simultaneous Equations? Simultaneous equations involve two or more equations with the same variables. The solutions to simultaneous equations are the values that, when substituted in for the variables, satisfy all of...

# Advanced Mathematics Support Programme ## Simultaneous Equations ### What are Simultaneous Equations? Simultaneous equations involve two or more equations with the same variables. The solutions to simultaneous equations are the values that, when substituted in for the variables, satisfy all of the equations. **Example**: $x + y = 5$ $x - y = 1$ The solution to this set of simultaneous equations is: $x = 3$, $y = 2$ ### Solving Simultaneous Equations Graphically Each equation can be plotted as a line on a graph. The solution to the simultaneous equations is the point where the lines intersect. **Example**: $y = x + 1$ $y = -x + 3$ These lines intersect at the point $(1, 2)$, so the solution is $x = 1$, $y = 2$. ### Solving Simultaneous Equations Algebraically #### 1. Elimination Rearrange one or both equations so that the coefficient of one of the variables is the same (but with opposite signs). Add the equations together to eliminate that variable. Solve the resulting equation. Substitute back into one of the original equations to find the value of the eliminated variable. **Example**: $2x + y = 7$ $x - y = -1$ Add the equations to eliminate $y$: $3x = 6$ $x = 2$ Substitute $x = 2$ into $x - y = -1$: $2 - y = -1$ $y = 3$ #### 2. Substitution Rearrange one of the equations to make one variable the subject. Substitute this into the other equation. Solve the resulting equation. Substitute back into the rearranged equation to find the value of the other variable. **Example**: $y = 2x + 1$ $3x + y = 6$ Substitute $y = 2x + 1$ into $3x + y = 6$: $3x + (2x + 1) = 6$ $5x + 1 = 6$ $5x = 5$ $x = 1$ Substitute $x = 1$ into $y = 2x + 1$: $y = 2(1) + 1$ $y = 3$ ### When to use each method - Elimination is best when the coefficients of one of the variables are already the same or easy to make the same. - Substitution is best when one of the equations is already in the form $y =$ something or $x =$ something ### Equations with more complex terms The elimination and substitution methods can also be used for simultaneous equations that involve more complex terms. **Example**: $x^2 + y^2 = 13$ $y = x + 1$ Substitute $y = x + 1$ into $x^2 + y^2 = 13$: $x^2 + (x + 1)^2 = 13$ $x^2 + x^2 + 2x + 1 = 13$ $2x^2 + 2x - 12 = 0$ $x^2 + x - 6 = 0$ $(x + 3)(x - 2) = 0$ $x = -3$ or $x = 2$ Substitute $x = -3$ into $y = x + 1$: $y = -3 + 1$ $y = -2$ Substitute $x = 2$ into $y = x + 1$: $y = 2 + 1$ $y = 3$ So the solutions are $x = -3$, $y = -2$ and $x = 2$, $y = 3$