Simultaneous Equations

Understanding Simultaneous Equations

• Simultaneous equations are a set of equations with multiple variables that are solved together, or simultaneously.
• These equations represent situations where multiple conditions are at play at the same time.
• The goal is to find the values of the variables that satisfy all of the equations in the set.
• There are different methods for solving simultaneous equations, including substitution, elimination and graphical methods.
• A solution to simultaneous equations is an ordered pair of numbers that makes both equations in the system true.

Solving Simultaneous Equations by Substitution

• Choose one of the equations and solve it for one variable, then substitute this into the other equation.
• After substituting, an equation in one variable is formed. Solve this equation to get the value of one of the variables.
• Substitute this value back into either of the original equations to find the value of the other variable.

Solving Simultaneous Equations by Elimination

• When the coefficients of one variable in both equations are the same, or can easily be made the same, you can use the elimination method.
• Multiply both equations, if necessary, so that the coefficients of one of the variables are the same.
• Then add or subtract the equations to eliminate one of the variables, leaving a simple linear equation to solve.
• Substitute the value of the single variable back into one of the original equations to find the other variable’s value.

Visualising Simultaneous Equations

• Another way to understand or solve simultaneous equations is to depict them as lines on a graph.
• Each equation is a line on the cartesian plane. The solution is the point where the lines intersect.
• By drawing the lines accurately, the solution can be found by looking at the point of intersection.

Common Mistakes with Simultaneous Equations

• A common trap when solving by substitution is to incorrectly distribute terms. Always make sure to distribute correctly when substituting an expression into another equation.
• Be mindful of signs when using the elimination method. The goal is to eliminate a variable, so strategically add or subtract the equations.
• When graphing, it is crucial to label the axes and the lines, to clearly identify which equation each line corresponds to. The intersection of lines should also be accurately defined.
• If answers do not seem logically consistent, check calculations through each step for possible mistakes. It’s important to always verify solutions by substituting them back into the original equations.