Math  /  Algebra

QuestionHow many solutions does the system of equations below have? 8x+3y=319x+5y=13\begin{array}{l} 8 x+3 y=3 \\ 19 x+5 y=-13 \end{array} no solution one solution infinitely many solutions

Studdy Solution

STEP 1

What is this asking? We've got two equations with two unknowns, xx and yy, and we need to figure out if they cross at a single point (one solution), never cross (no solutions), or are the same line (infinitely many solutions)! Watch out! Don't just guess based on how the equations look!
We need to do a little math to figure out what's really going on.

STEP 2

1. Elimination Setup
2. Solve for yy
3. Solve for xx
4. Verify Solution

STEP 3

Alright, let's **get rid of** yy using elimination!
We've got 3y3y in the first equation and 5y5y in the second equation.
To eliminate yy, we need to make the coefficients of yy match but with opposite signs.

STEP 4

Let's multiply the first equation by **5** and the second equation by **-3**.
This gives us: 5(8x+3y)=535 \cdot (8x + 3y) = 5 \cdot 3 40x+15y=1540x + 15y = 15and 3(19x+5y)=3(13)-3 \cdot (19x + 5y) = -3 \cdot (-13) 57x15y=39-57x - 15y = 39

STEP 5

Now, let's **add those two new equations** together: (40x+15y)+(57x15y)=15+39(40x + 15y) + (-57x - 15y) = 15 + 39 17x=54-17x = 54x=5417x = -\frac{54}{17}Whoa, a fraction!
Don't worry, fractions are friends!

STEP 6

Now, let's **plug this value of** xx back into the first original equation, 8x+3y=38x + 3y = 3: 8(5417)+3y=38 \cdot \left(-\frac{54}{17}\right) + 3y = 3 43217+3y=3-\frac{432}{17} + 3y = 33y=3+432173y = 3 + \frac{432}{17}3y=5117+432173y = \frac{51}{17} + \frac{432}{17}3y=483173y = \frac{483}{17}y=1348317y = \frac{1}{3} \cdot \frac{483}{17}y=16117y = \frac{161}{17}

STEP 7

We already found xx in the process of solving for yy!
Remember, we got x=5417x = -\frac{54}{17}.

STEP 8

Let's **double-check** our work by plugging our values for xx and yy into the *second* original equation, 19x+5y=1319x + 5y = -13: 19(5417)+5(16117)=1319 \cdot \left(-\frac{54}{17}\right) + 5 \cdot \left(\frac{161}{17}\right) = -13 102617+80517=13-\frac{1026}{17} + \frac{805}{17} = -1322117=13-\frac{221}{17} = -1313=13-13 = -13Boom! It works!

STEP 9

We found a single, unique solution for both xx and yy, so the system of equations has *one solution*.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord