Simultaneous Equations - Year 11 PDF Download

CIE IGCSE Maths: Extended

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First teaching 2023

First exams 2025

Simultaneous Equations

Linear Simultaneous Equations

• When we have two unknowns, such as x and y, we need two equations to find both simultaneously. These equations are known as simultaneous equations. By solving these equations, we can determine the values of both x and y. For instance, if we have 3x - 2y = 11 and 2x - y = 5, the solutions would be x = 3 and y = 1. Linear simultaneous equations contain only x and y terms, without any higher powers or products involving x and y.

• To find the values of x and y in simultaneous equations, like 3x - 2y = 11 and 2x - y = 5, we solve the equations simultaneously to obtain x = 3 and y = 1.

• Solving simultaneous equations involves determining the values of two unknowns, x and y, by working with two equations simultaneously. For example, if we have 3x - 2y = 11 and 2x - y = 5, the solutions would be x = 3 and y = 1.

Linear Simultaneous Equations and Elimination Method

Linear simultaneous equations involve two or more equations with the same variables. The elimination method is a systematic way to solve such equations by eliminating one variable.

Understanding Linear Simultaneous Equations

• Linear simultaneous equations contain variables like x and y, without higher powers or products between them.
• For example, in the equations 3x + 2y = 11 and 2x - y = 5, x = 3 and y = 1 are the solutions.

Solving Linear Simultaneous Equations by Elimination

The elimination method involves eliminating one variable to solve the equations. Here's how it works:

• Eliminate x's from 3x + 2y = 11 and 2x - y = 5:
  • Multiply every term in the first equation by 2 to get 6x + 4y = 22.
  • Multiply every term in the second equation by 3 to get 6x - 3y = 15.
  • Subtract the second equation from the first to eliminate 6x, resulting in 4y - (-3y) = 22 - 15, simplifying to 7y = 7.
  • Solve for y (y = 1) and substitute it back to find x (x = 3).
• Eliminate y's from 3x + 2y = 11 and 2x - y = 5:
  • Multiply every term in the second equation by 2 to get 4x - 2y = 10.
  • Add this to the first equation to eliminate 2y, continuing the process.
  • Check the final solutions by verifying they satisfy both equations.

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How to Solve Linear Simultaneous Equations by Substitution

• Multiply every term in the first equation by 2: 6x + 4y = 22
• Multiply every term in the second equation by 3: 6x - 3y = 15
• Subtract the second result from the first to eliminate the 6x's, leaving 4y - (-3y) = 22 - 15, i.e., 7y = 7
• Solve to find y (y = 1) then substitute y = 1 back into either original equation to find x (x = 3)

Alternatively

• To eliminate the y's from 3x + 2y = 11 and 2x - y = 5:
• Multiply every term in the second equation by 2: 4x - 2y = 10
• Add this result to the first equation to eliminate the 2y's (as 2y - 2y = 0). The process then continues as above

• Check your final solutions satisfy both equations

How to Solve Linear Simultaneous Equations by Substitution

• When faced with a system of linear equations, such as 6x + 4y = 22 and 6x - 3y = 15, one method of solving involves substitution.
• Begin by isolating one variable in one of the equations; for example, solving for y in the first equation yields y = 1.
• Substitute this value back into either original equation to solve for the other variable; in this case, x = 3.

Alternate Approach

• If faced with equations like 3x + 2y = 11 and 2x - y = 5, you can eliminate a variable by multiplying terms and adding equations together.
• Multiplying the second equation by 2 gives 4x - 2y = 10. Adding this to the first equation cancels out the y terms, allowing you to solve for x and then y.

• Always verify your solutions by plugging them back into both original equations to ensure they satisfy all conditions.

Linear Simultaneous Equations and Substitution

"Substitution" involves substituting one equation into the other to solve for variables.

• Rearrange one of the equations into y = ... (or x = ...).
• For example, if we have 3x - 2y = 11 and 2x - y = 5, we can rearrange the second equation to y = 2x - 5.
• Substitute the rearranged equation into the first equation (replace all y's with 2x - 5).
• 3x - 2(2x - 5) = 11
• Solve this equation to find x (x = 3), then substitute x = 3 into y = 2x - 5 to find y (y = 1).
• Check your final solutions to ensure they satisfy both equations.

Using Graphs to Solve Linear Simultaneous Equations

Graphs can be a helpful visual tool to solve systems of linear equations.

• Graph each equation on the same set of axes.
• The point of intersection of the graphs represents the solution to the system.
• If the lines are parallel, there is no solution as they do not intersect.
• If the lines overlap, there are infinite solutions as all points on the line are valid solutions.

Plotting and Solving Linear Simultaneous Equations Graphically

• Plot both equations on the same set of axes. You can use a table of values or rearrange into y = mx + c if that helps.
• Find where the lines intersect (cross over). The x and y solutions to the simultaneous equations are the x and y coordinates of the point of intersection.
• For example, to solve 2x - y = 3 and 3x - y = 4 simultaneously, first plot them both and find the point of intersection (2, 1). The solution is x = 2 and y = 1.

How to Solve Linear Simultaneous Equations from Worded Contexts

• Always check that your final solutions satisfy the original simultaneous equations. This helps you confirm if you have the correct solutions or not.

Solve the simultaneous equations.

Solving Simultaneous Equations

In this lesson, we will discuss how to solve a system of simultaneous equations using a step-by-step approach.

Step 1: Number the Equations

• Assign numbers to each equation in the system for easy reference.

Step 2: Make the y Terms Equal

• To make the y terms equal, multiply all parts of one equation by a constant to match the coefficients.
• This step helps in eliminating the y variable when the equations are added or subtracted.
• For example, if you have 5x - 2y = 11 and 4x - 3y = 18, you can multiply the first equation by 3 and the second by 2 to proceed.

Step 3: Eliminate the y Terms

• Add or subtract the modified equations to eliminate the y variable and solve for x.
• After making the y terms equal, you can combine the equations to cancel out the y variable.

Step 4: Solve for Variables

• Once one variable is determined, substitute its value back into one of the original equations to find the other variable.
• Continue solving until you have values for both x and y that satisfy both equations simultaneously.

Solving Equations with Substitution

• Substitute into either of the two original equations.

Solving for y

• Solve this equation to find y.

Verification of Solutions

• Substitute x = 3 and y = -2 into the other equation to check that they are correct.

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Quadratic Simultaneous Equations

• When there are two unknowns (say x and y) in a problem, we need two equations to be able to find them both: these are called simultaneous equations

• If there is an x^2 or y^2 or xy in one of the equations, then they are quadratic (or non-linear) simultaneous equations

• Identify the quadratic simultaneous equations in the problem.
• Write down the two equations involving the unknowns x and y.
• Eliminate one variable (either x or y) by adding or subtracting the equations.
• Solve the resulting linear equation to find one of the variables.
• Substitute the value found back into one of the original equations to solve for the other variable.
• Check the solutions by plugging them back into both equations to ensure they satisfy both equations.

Method of Substitution in Solving Equations

• Utilize the method of substitution by replacing variables in equations
• Avoid substituting complex equations into simpler ones

Step-by-Step Process

• Begin by solving x^2 - y^2 = 25 and y - 2x = 5
• Rearrange the linear equation as y = 2x - 5
• Substitute y = 2x - 5 into the quadratic equation
• Expand and solve the resulting quadratic equation to find x = 0 and x = -4
• Substitute each x-value back into y = 2x - 5 to determine y
• Present solutions in an organized manner: (x = 0, y = 5) or (x = -4, y = -3)

Importance of Correct Substitution

• Incorrect substitutions can lead to erroneous solutions
• Proper substitution ensures accurate results in mathematical equations

Illustrative Example

• Consider the equation x^2 - y^2 = 25 and y - 2x = 5
• Substitute y = 2x - 5 into x^2 - (2x - 5)^2 = 25
• Incorrect substitution might yield incorrect values of x and y

How to Solve Quadratic Simultaneous Equations

• Rearrange the linear equation into y = 2x - 5
• Substitute this into the quadratic equation, replacing all y's with (2x - 5) in brackets: x^2 - (2x - 5)^2 = 25
• Expand and solve this quadratic equation to find x = 0 and x = -4
• Substitute each value of x into the linear equation, y = 2x - 5, to find the corresponding values of y
• Present the solutions clearly: x = 0, y = 5 or x = -4, y = -3

Check Solutions

• Check that the final solutions satisfy both equations

Using Graphs to Solve Quadratic Simultaneous Equations

Graphs can be helpful in visualizing and solving quadratic simultaneous equations. By plotting the graphs of the two equations, the points of intersection represent the solutions to the system of equations.

Here's how you can use graphs to solve quadratic simultaneous equations:

• Plot the graphs of both equations on the same coordinate system
• The points where the graphs intersect are the solutions to the system of equations

• Plotting Equations on a Graph:
• To plot two equations on the same graph, you can utilize a table of values or rearrange them into the form y = mx + c, especially for straight lines.
• Finding Intersection Points:
• Determine where the lines intersect, indicating the solutions to the simultaneous equations at the x and y coordinates of the point of intersection.
• Example of Solving Simultaneous Equations:
• For instance, if you need to solve y = x^2 - 3x - 1 and y = 2x - 1 at the same time, plot both equations and identify the points of intersection at (-1, -1) and (0, 1). The solutions are x = -1, y = -1 or x = 0, y = 1.

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Understanding Quadratic Equations and Tangents

• When a quadratic equation has a repeated root, it signifies that the corresponding line is a tangent to the curve. On the other hand, if the quadratic equation has no roots, this indicates that the line does not intersect with the curve, or an error has been made in the calculations.
• It's crucial to explicitly pair the x and y values in your final answer. Failing to do so could lead to losing marks, even if the solution is otherwise correct.
• It's a common error to mistakenly believe that each squared term in an equation like x² + y² = 25 can be square-rooted to yield x + y = 5. Remember, this assumption is incorrect. The most simplification possible is √25, but it's advised not to isolate x or y in this context.

Problem Solving: Equations to Solve

Solve the following system of equations:

x² - y² = 36

x = 2y - 6

Strategy for Solving

Given that there is one quadratic equation and one linear equation, the solution involves substitution.

Equation (2) can be eliminated by substituting it into Equation (1).

Substitute the expression from Equation (2) into Equation (1).

Step-by-Step Solution

Follow these steps to derive the solution:

Algebraic Expressions and Equations

• Expand the brackets by treating squared brackets like double brackets.
• Simplify the given expression.
• Rearrange the expression to form a quadratic equation equal to zero.
• Factorize the expression by taking out the common factor.
• 

Finding Solutions

• Solve the equation to find the values of the variable.Let each factor be equal to 0 and solve.
• Let each factor be equal to 0 and solve.
• Substitute the values of the variable into one of the equations to find the values of another variable.

Substitute Equations

• Substitute the values into one of the equations (the linear equation is easier) to find the values of .

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