Question

1.How do you use substitution to solve a system of linear equations?

3.Durante solves a system of equations, in x and y, by substitution and gets an answer of 5 = 5. He states that the solution to the system is all of the points (x,y where x and y are real numbers. Describe Duante's error.

4.When solving a system of equations using substitution, how can you determine whether the system has one solution, no solution, or infinitely many solutions?

5. y = 6 - x
4x - 3y = -4

6. x = -y + 3
3x - 2y = -1

Answer 1

1. How to use substitution to solve a system of linear equations (step-by-step)
- Solve one equation for one variable in terms of the other (e.g., get y = ... or x = ...).
- Substitute that expression into the other equation so it has only one variable.
- Solve that single-variable equation.
- Substitute that value back into the expression from step 1 to get the other variable.
- Check the pair (x,y) in both original equations.

(Geometric meaning: substitution finds the intersection point of the two lines.)

3. Describe Duante's error
- If substitution leads to a true statement with no variables (e.g., 5 = 5), that means the two equations are dependent — they represent the same line — so there are infinitely many solutions, but NOT every pair of real numbers. The solution set is all points (x,y) on that line, not all real ordered pairs. Duante incorrectly concluded "all (x,y) with x and y real" (the entire plane).

4. How to tell whether a system has 1, 0, or infinitely many solutions by substitution
- One solution: after substitution you get a specific value for the variable (e.g., x = 2), leading to one (x,y) pair. Geometrically: the lines intersect at one point.
- No solution: you get a false statement with no variables (e.g., 0 = 5). Geometrically: the lines are parallel distinct lines and never meet.
- Infinitely many solutions: you get a true identity with no variables (e.g., 0 = 0 or 5 = 5). Geometrically: the two equations represent the same line.

5. Solve
y = 6 - x
4x - 3y = -4

Substitute y:
4x - 3(6 - x) = -4
4x - 18 + 3x = -4
7x - 18 = -4
7x = 14
x = 2
y = 6 - 2 = 4

Solution: (2, 4)

6. Solve
x = -y + 3 (same as x = 3 - y)
3x - 2y = -1

Substitute x:
3(3 - y) - 2y = -1
9 - 3y - 2y = -1
9 - 5y = -1
-5y = -10
y = 2
x = 3 - 2 = 1

Solution: (1, 2)