2.2.3 Solving equations algebraically, graphically

• What are simultaneous equations used to find?

  Common values for variables
• To eliminate a variable, its coefficients must be equal or opposite.

  True
• Steps for solving simultaneous equations by elimination

  1️⃣ Line up variables vertically

  2️⃣ Make coefficients of one variable equal/opposite

  3️⃣ Add equations to eliminate one variable

  4️⃣ Solve for remaining variable

  5️⃣ Substitute back into original equation

  6️⃣ Solve for second variable

  7️⃣ Check solution
• How does the solution $x =$$2, y =$$4$ satisfy both of the original simultaneous equations?


  Substituting $x =$$2, y =$$4$ into both equations gives a true statement
• Why do you need to make the coefficients of the variables you want to eliminate opposite when solving simultaneous equations by elimination?

  So that the variables cancel out when you add the equations
• If you want to eliminate the variable x, which coefficients would you make opposite?

  2 and 5
• After eliminating one variable, what should you solve for?

  Remaining variable
• What are the steps to solve simultaneous equations by elimination?

  1. Line up the variables
  2. Determine which variable to eliminate, make the coefficients opposites
  3. Add straight down (one variable should be eliminated)
  4. Solve the resulting equation
  5. Substitute the result back into one of the original equations to find the ordered pair solution
• What is the fourth step in the substitution method?

  Substitute back into the equation
• Simultaneous equations require finding variable values that satisfy all equations at the same time.


  True
• What is the first step in solving simultaneous equations by elimination?

  Line up the variables
• What is the final solution to the simultaneous equations?

  • $x =$$2$
  • $y =$$4$
• What is the substitution method used for?

  Solving simultaneous equations
• Checking the solution involves substituting the values of $x$ and y</latex> into both original equations


  True
• Substitution is used to find the value of the second variable.


  True
• Steps of the substitution method

  1️⃣ Rearrange one equation to isolate one variable

  2️⃣ Substitute this expression into the other equation

  3️⃣ Solve for the remaining variable

  4️⃣ Substitute back into the rearranged equation

  5️⃣ Check the solution in both original equations
• The equation $5 - 2y =$$1$ simplifies to y = 2</latex>


  True
• What does the point of intersection (5, -2) represent?

  The point of intersection (5, -2) represents the solution to both equations simultaneously.
• What is the final step in solving simultaneous equations by elimination?

  Check solution
• What is the x-coordinate of the point of intersection?

  5
• What does each equation represent when solving simultaneous equations graphically?

  A straight line
• Match the steps of the substitution method with their descriptions:

  Rearrange ↔️ Isolate one variable

  Substitute ↔️ Replace a variable with its expression

  Solve ↔️ Find the value of the variable

  Check ↔️ Verify the solution
• How can the equations of the two lines be used to find the point of intersection?

  • Set the two equations equal to each other and solve for the x-coordinate
  • Substitute the x-coordinate back into one of the original equations to find the y-coordinate
• The lines in the example intersect at (5, -2), meaning x = 5 and y = -2 is the solution.


  True
• What is the value of y at the intersection point in the example?

  -2
• What is the second step in the substitution method?

  Substitute the expression
• What is the third step in the substitution method?

  Solve for the variable
• What is the y-coordinate of the point of intersection?

  -2
• What is the equation for the first line shown in the image?

  x - y = 1
• How can the point of intersection be used to solve the system of equations?

  • The point of intersection (5, -2) satisfies both equations simultaneously
  • This means the solution to the system of equations is x = 5 and y = -2
• What is the point of intersection of the two lines shown in the image?

  (5, -2)
• What is the value of x at the intersection point in the example?

  5
• From the equation x + y = 5</latex>, we can isolate $x$ as $x =$$5 - y$


  True
• What is the equation of the first line shown in the image?

  $y =$$-2x +$$8$
• How can you use the graphing method to solve a system of linear equations?

  To solve a system of linear equations using the graphing method:

  1. Graph each equation on the same coordinate plane
  2. The point where the lines intersect is the solution to the system
• What is the equation for the second line shown in the image?

  2x + y = 8
• What is the equation of the second line shown in the image?

  $y =$$\frac{4}{5}x - 6$
• What are the key steps in the graphing method for solving a system of linear equations?

  Key steps in the graphing method:

  1. Plot the lines represented by each equation
  2. Identify the point where the lines intersect
  3. The coordinates of the intersection point are the solution to the system
• What is the equation for the fourth line shown in the image?

  y = -2x + 8
• What is the value of x after solving the equation -2x + (x - 1) = 8?


  3