Question

Use algebra to find the point at which the line

Answer 1

`x=-(19)/(3)\\\\ y=-(9)/(2)`

Explanation:

• Because the two equations are both in the slope-intercept form (y = mx + b, where m is the slope and b is the y-intercept), we can first find x by setting the two equations equal to each other:

g(x) = h(x)

-7/4x -187/12 = 9/5x + 69/10

Step 1: Add -187/12 to both sides of the equation

(-7/4x -187/12 = 9/5x + 69/10) + 187/12

-7/4x = 9/5x + 1349/60

Step 2: Subtract 9/5x from both sides of the equation

(-7/4x = 9/5x + 1349/60) - 9/5x

-71/20x = 1349/60

Step 3: Divide both sides of the equation by -71/20 to solve for x

(-71/20x = 1349/60) / -71/29

x = -19/3

• Now that we've found the x-coordinate of the intersection point, we can plug in -19/3 for x into any of the two equations to find the y-coordinate of the intersection point
• Let's try the g(x)

g(-19/3) = -7/4(-19/3) -187/12

g(-19/3) = 133/12 - 187/12

g(-9/2) = -9/2

Thus the point (x, y) at which they intersect is (-19/3, -9/2)

Optional Final Step

• We can check that the points we found are the intersection point by plugging in -19/3 for x in both equations and seeing whether the result we get is -9/2 for both equations

g(-19/3)

-9/2 =-7/4(-19/3) - 187/12

-9/2 = 133/12 - 187/12

-9/2 = -9/2

h(-19/3)

-9/2 = 9/5(-19/3) + 69/10

-9/2 = -57/5 + 69/10

-9/2 = -9/2

Answer 2

• x = -19/3
• y = -9/2

Explanation:

You want the solution g(x) = h(x) to the system of equations ...

• g(x) = -7/4x -187/12
• h(x) = 9/5x +69/10

Standard form

Using y for g(x) and h(x), we can multiply the first equation by 12 and add 21x to put it in standard form:

21x +12y = -187

Similarly, we can multiply the second equation by 10 and subtract (10y+69) to get ...

18x -10y = -69

Elimination

Adding 5 times the first equation to 6 times the second gives ...

5(21x +12y) +6(18x -10y) = 5(-187) +6(-69)

213x = -1349 . . . . . . . . . . . simplify. Note that y has been eliminated.

x = -1349/213 = -19/3 . . . . divide by the x-coefficient

Substituting for x in the second of our rewritten equations, we get ...

18(-19/3) -10y = -69

-10y = 45 . . . . . . . . . . add 114

y = -9/2 . . . . . . . . . . divide by the y-coefficient

The solution is ...

• x = -19/3
• y = -9/2