Solve the system of linear equations by graphing.

answers:

(-5,0)

(-3,4)

(4,5)

(-1,0)

(0,6)

(1,2)

(-1,-2)

! i don’t understand this problem. can someone explain it to me? you! (questions 5-10)

5. (-1, 0)

6. (-1, -2)

7. (-3, 4)

8. (1, 2)

9. (-5, 0)

10. (0, 6)

11. (4, 5) . . . . . the only remaining answer, which also happens to be correct

Step-by-step explanation:

Each of the equations is the equation for a line. That is, the graph of it is a straight line. Every pair of coordinates on that line is a solution to the equation.

When equations for two lines with different slopes are graphed, the lines will intersect. The coordinates of the point of intersection will satisfy both equations, so are a solution to that pair of equations.

When the problem asks you to "solve the system of linear equations by graphing", it is asking you to graph each of the equations in the pair, and report the (x, y) coordinates of the point where the lines intersect.

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There are different usable methods for graphing equations depending on the form in which they are presented.

Slope-intercept form

Here the equations for the first three problems are in "slope-intercept" form. That means the constant in the equation tells you where the line crosses the y-axis, the y-intercept. The coefficient of x in each equation is the slope. It tells you the ratio of rise to run. A slope of -2 (as in the first equation of problem 6) tells you the graph of than line falls 2 units (negative 2 units of rise) for each unit it goes to the right (1 unit of run). That equation has a y-intercept of -4, so another point on that line is (1, -6), which is 1 unit right and 2 units down from the y-intercept at (0, -4).

Standard form

The equation for the last three problems are in "standard" form (except for the 2nd equation of problem 9). An equation in this form can be graphed fairly easily by putting it into "intercept form." (Not slope-intercept form.) An equation in intercept form looks like ...

  x/(x-intercept) + y/(y-intercept) = 1

You generally get to this form by dividing by the constant and expressing the x- and y-coefficients as denominators. Here's an example.

Problem 8: 4x/2 -y/2 = 1 ⇒ x/(1/2) +y/(-2) = 1

  This means the x-intercept of this equation is (1/2, 0) and the y-intercept is (0, -2). (See the solid red line in the second attachment for a graph of this.)

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About the graphs

The first equation of each pair is graphed as a solid line; the second is graphed as a dashed line. For inequalities, the dashed line has a particular meaning. Here, it just means that is the graph of the second equation. Colors of the lines for the two equations are the same. Equations for problems 5, 6, and 7 are graphed in red, blue, and green in the first attachment. Equations for problems 8, 9, and 10 are graphed in red, blue, and green in the second attachment. In each case, the intersection point (solution to the system) has its coordinates shown.

gdp is the best phrase!

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