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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Solving Systems of Equations - Two Lines

Intersecting lines should be solved algebraically, even if the directions say to solve graphically.

⇒ Recall: Two algebraic methods for solving systems of equations are addition and substitution.

Example 1:  Check:

adding these equations gives 3x = 6 which yields x = 2.

Substituting in equation 2: (2) - y = 1 which yields y = 1.

The solution is the point (2,1) or S = { (2,1) }.

Example 2:  Check:

Multiple the second equation by 3 and obtain: Adding these equations gives 11x = 22 or x = 2.

Substitute in second equation: 3(2) - y = 1 or y = 5.

Check by substituting both in first equation: 2(2) + 3(5) = 19 Thus, the solution is the point (2,5) or S = { (2,5) }.

Example 3:  The solution is the point (2, 3) or { (2, 3) }

Since the second equation is already solved for y use the expression as a replacement for y in the first equation:

x + (2x − 1) = 5 which gives us an equation in x alone.

3x = 6 or x = 2. Substitute in the second equation to solve for y.

y = 2(2) − 1 or y = 3. To check substitute both in the first equation: 4 + 7 = 5 Example 4:  The solution is the point (4, 2) or { (4, 2) }

Since the second equation is already solved for y use the expression as a replacement for y in the first equation:

2Â·(-2y + 8) − 3y = 2 which gives us an equation in y alone.

- 7y = -14 or y = 2. Substitute in the second equation to solve for x.

x = -2(2) + 8 or x = 4. To check substitute both in the first equation: 8 − 6 = 2 