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## Algebra basics

### Course: Algebra basics > Unit 4

Lesson 2: Solutions to two-variable linear equations# Worked example: solutions to 2-variable equations

How do you check if an ordered pair is a solution to a given equation? You need to plug in the numbers and see what equality results. Watch this video to see a worked example.

## Want to join the conversation?

- At0:33, would the equation y = -2+ 4/3y be equivalent?(14 votes)
- where did this dude go to school. Mans smart(11 votes)
- Massachusetts Institute of Technology (MIT) and University of New Orleans.(5 votes)

- what if you are not given an ordered pair and you have to figure out this equation(5 votes)
- How are you going to create an equation if you are not given at least one ordered pair? The ordered pair could be implied by giving the x or y intercept, but it is still an ordered pair.(7 votes)

- does this cover absolute vale equalities and inequalities(4 votes)
- I saw nothing here about absolute values which would look like "|x|" or |3|=3 or |-3|=3 OR |-9|=9

The best way to think about absolute values is- "No matter the sign (+ or -) the number remains positive, because the absolute value cannot be negative" Hope that helps. Now inequalities, something simple like....... 1<x<3 where "x" is greater then 1 but less then 3.. The interval notation would be (SET BUILDER NOTATION!)----> {x|x>1, x<3} or (INTERVAL NOTATION(WHICH IS INCLUDING A UNION aka "U"))--> (1,x)U(x,3)...... if the sign was a "greater than or equal to, or less than or equal to" then the interval notation would be [1,x]U[x,3].... I hope this helps.(8 votes)

- What is an "ordered" pair?(2 votes)
- A point on the coordinate plane that gives the horizontal distance from the origin (x) and the vertical distance (y) that is in the form (x,y). If you create a table from a linear equation, you get a series of ordered pairs.(10 votes)

- can't you simplify the equation first? like 4x + 3y = 6?(3 votes)
- You sure can - if you think that will help you.

Look what happened in your case: 4x - 1 = 3y + 5 simplifies to 4x - 3y = 6**not 4x + 3y = 6**

Keep Studying!

Keep Asking Questions!(5 votes)

- how do you figure the value of x and y(2 votes)
- You can pick
**any**number to use for one of the variables. Plug it into the equation to calculate the other variable. For example: 2x+3y=12

If x=2: 2(2)+3y = 12

-- Simplify: 4+3y = 12

-- Subtract 4 from both sides: 3y = 8

-- Divide both sides by 3: y = 8/3 or 2 2/3

-- You now have a point on the line: (2, 2 2/3)

In the video, Sal is given points to test. In that situation, you replace each variable with their given value and see if the 2 sides are equal. Remember, ordered pairs are always (x,y). So, the first value is X and the 2nd value is Y.

Hope this helps.(7 votes)

- between x and y what is the slope and the intercept?(2 votes)
- Most lines have 2 intercepts: an x-intercept and a y-intercept. The x-intercept occurs where y=0 which places the point on the x-axis. The y-intercept occurs where x=0 which places the points on the y-axis.

Lines that have only one intercept are: a) Lines that cross both the x & y axis at the origin (0,0), and b) vertical and horizontal lines which only cross one axis.

To find the slope of a line, you need two points that are on the line, then you would use the slope formula: m = (y2-y1)/(x2-x1). Or, if your equation is in slope intercept form (y = mx + b), then you can read the slope from the line. The number (m) that is the coefficient of "x" is the slope. And, the constant term (b) is the y-intercept at (0,b).(7 votes)

- solve the following systems of linear equations by elimination

12x+5y=23

2x-7y=39(4 votes)- Add two times the first equation to the second equation to eliminate x:

12x + 5y = 23

2x - 7y = 39

(12x + 5y) + 2(2x - 7y) = 23 + 2 * 39

26x - 18y = 101

So, y = (26x - 101) / -18. Substitute the expression for y back into the first equation to solve for x:

12x + 5y = 23

12x + 5((26x - 101) / -18) = 23

Expand the right side and simplify:

12x + (130x - 505) / -18 = 23

-6x - 505 / -18 = 23

Multiply both sides by -18 to get rid of the fraction:

6x + 505 = -414

Subtract 505 from both sides to isolate x:

6x = -919

Divide both sides by 6 to find x:

x = -153

Finally, substitute the value of x back into the expression for y:

y = (26x - 101) / -18

y = (26 * -153 - 101) / -18

y = 24.

The solution is x = -153 and y = 24.(2 votes)

- So my math book says that I should rewrite the equation first, simplify it, then graph it. how do I rewrite it? I keep trying and I keep getting the wrong answer.

P.s. It gives me stuff like, 4X + Y = 6(4 votes)- rewriting for graphing purposes usually means to put in slope intercept form (isolate the y).

Example: 4x + 3y = 9. Subtract 4x to get 3y=-4x+9

Divide all by 3 to get 3/3 y= -4/3 x + 9/3, reduce to get y=-4/3 x + 3. Graph y intercept at (0,3), go down 4 right 3 to get to point (3,-1).(1 vote)

## Video transcript

- [Voiceover] "Which of
the ordered pairs is a "solution of the following equation?" 4x minus one is equal to 3y plus five. Now, when we look at an
ordered pair we wanna figure out whether it's a
solution, we just have to remind ourselves that
in these ordered pairs the convention, the standard,
is is that the first coordinate is the x coordinate,
and the second coordinate is the y coordinate. So they're gonna, if this is
a solution, if this ordered pair is a solution, that means
that if x is equal to three and y is equal to two,
that that would satisfy this equation up here. So let's try that out. So, we have four times x. Well we're saying x needs
to be equal to three, minus one, is going to be
equal to three times y. Well, if this ordered
pair is a solution then y is going to be equal
to two, so three times y, y is two, plus five. Notice all I did is wherever I saw the x, I substituted it with three, wherever I saw the y, I
substituted it with two. Now let's see if this is true. Four times three is twelve, minus one. Is this really the same
thing as three times two which is six, plus five? See, 12 minus one is 11, six plus five is also 11. This is true, 11 equals 11. This pair three, two does
satisfy this equation. Now let's see whether
this one does, two, three. So this is saying when x is equal to two, y would be equal to
three for this equation. Let's see if that's true. So four times x, we're
now gonna see if when x is two, y can be three. So four times x, four times two, minus one is equal to three times y, now y we're testing to
see if it can be three. Three times three plus five, let's see if this is true. Four times two is eight, minus one, is this equal to three times three? So that's nine plus five. So is seven equal to 14? No, clearly seven is not equal to 14. So these things are not
equal to each other. So this is not a solution,
when x equals two y cannot be to three and
satisfy this equation. So only three, two is a solution.