(1) We are asked to find the point of intersection of the lines y=3x+4 and y=-2x+9.

There are several methods to find this point, assuming it exists. Given the equations of two lines, the graphs of the lines could be parallel (thus no intersection point), intersecting (resulting in a single intersection point), or the same.

Solving for the intersection of lines is an example of solving a system of linear equations. If the lines are parallel, the system is said to be inconsistent. If the system is consistent then either the equations describe the same line (a dependent set) or the lines intersect (an independent set.)

Some methods to solve are to graph the lines, solve using substitution, using linear combinations, using Cramer's method, using inverse matrices, setting up an augmented matrix and manipulating into reduced row echelon form, or even a guess and check strategy.

(a) We could graph the two lines. (See attachment.) **It appears that the point of intersection is (1,7)**. Substituting into each equation we see that 7=3(1)+4 and 7=-2(1)+9 are both true.

(b) Use substitution. Since y=3x+4, we can use the expression 3x+4 in place of y in the other equation. 3x+4=-2x+9. Adding 2x to both sides of the equation and subtracting 4 from both sides of the equation yields x=1. Then y=3(1)+4=7. Thus the solution is (1,7).

(c) Use linear combinations.

y=3x+4

y=-2x+9

Subtract the first equation from the second equation to get 0=-5x+5 or 5x=5 and x=1. If x=1 the y=-2(1)+9=7 and the solution is (1,7)

The other methods require some knowledge of matrices.

(2) You should try the second system on your own. If the two equations are y=-4x-7 and -y=5x+10 the solution is (-3,5).

**Further Reading**

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