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Matrices

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Matrices

- Solve for one variable in one of the equations.
- Substitute this expression into the other equation to get one equation with one unknown.
- Back substitute the value found in step 2 into the expression from step 1.
- Check.

Solve:

Step 1: Solve equation 1 for y.

Step 2: Substitute this expression into eqn 2 for y.

Step 3: Solve for x.

Step 4: Substitute this value for x into eqn 1 and find the corresponding y value.

Step 5: Check in both equations.

- Adjust the coefficients.
- Add the equations to eliminate one of the variable. Then solve for the remaining variable.
- Back substitute the value found in step 2 into one of the original equations to solve for the other variable.
- Check.

Solve:

Step1: The coefficients on the y variables are opposites. No multiplication is needed.

Step 2: Add the two equations together.

Step 3: Solve for x.

Step 4: Substitute this value for x into either eqn and find the corresponding y value.

Step 5: Check in both equations.

- In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
- In simple terms, a table of numbers.

- John and Bill went to the tuck shop. John bought a pie and cake and Bill bought 2 pies, a drink and a cake.

Matrix

- Two matrices are equal only if they have the same order and their corresponding elements are equal.

- Two matrices must be of the same order before they can be added.

- The negative matrix is obtained by taking the opposite value of each element in the matrix.

Subtraction is the same as adding the negative matrix.

- Multiplying by a number.
- Each element is multiplied by the number.
- Also

- Every element in the matrix is zero.
- This is the identity for addition.
- Example:

- The elements of the leading diagonal are all 1 and all other elements are zero.
- Example:

- Two matrices can be multiplied if the number of columns of the first matrix equals the number of rows of the second matrix.
- Example:

- This forms a 2 x 1 matrix

- This forms a 2 x 1 matrix

- A linear equation with 2 variables is a line
- E.g.

- Scenario 1
- Two lines intersect at one point. There is a unique solution.
- This means the system is independent and consistent with a uniquesolution.

- Two small jugs and one large jug can hold 8 cups of water. One large jug minus one small jug constitutes 2 cups of water. How many cups of water can each jug hold?

- Two small jugs and one large jug can hold 8 cups of water. One large jug minus one small jug constitutes 2 cups of water. How many cups of water can each jug hold?
- Let x = number of cups of water that the large jug holds
- Let y = number of cups of water that the small jug holds

- Two small jugs and one large jug can hold 8 cups of water. One large jug minus one small jug constitutes 2 cups of water. How many cups of water can each jug hold?
- Let x = number of cups of water that the small jug holds
- Let y = number of cups of water that the large jug holds

- Let x = number of cups of water that the small jug holds
- Let y = number of cups of water that the large jug holds

- Let x = number of cups of water that the small jug holds
- Let y = number of cups of water that the large jug holds

- This means the system is independent and consistent with a uniquesolution.
- Geometrically: The lines intersect at a unique point because the gradients are not the same.

- Solve the following equation:

- Solve the following equation:

- Solve the following equation:

- Solve the following equation:

- This means the system is independent and consistent with a uniquesolution.

- The two lines don’t intersect because the lines are parallel.
- The system is “independentand inconsistent and has nosolution.”

- Solve the following

Identical lines which intersect at an infinite number of points.

The system is “dependent and consistent and has an infinite number of solutions.”

- Solve the following