Question

svlc algebra 1a - standard (15260)
solving systems: introduction to linear combinations
what is the solution (q, r) to this system of linear equations?
12q + 3r = 15
-4q - 4r = -44
(-2, 13) (15, -44)
(8, -1) (-18, 29)

Explanation:

Step1: Simplify the second equation

Multiply the second equation \(-4q - 4r = -44\) by 3 to make the coefficients of \(q\) in both equations have a relationship for elimination. So we get \(-12q - 12r = -132\).

Step2: Add the two equations

Now add the first equation \(12q + 3r = 15\) and the new second equation \(-12q - 12r = -132\). \((12q - 12q)+(3r - 12r)=15 - 132\), which simplifies to \(-9r=-117\).

Step3: Solve for \(r\)

Divide both sides of \(-9r = -117\) by \(-9\). So \(r=\frac{-117}{-9}=13\). Wait, that seems wrong. Wait, no, let's check again. Wait, maybe I made a mistake in the multiplication. Wait, let's try another way. Let's solve the second equation for \(q\). From \(-4q - 4r=-44\), divide both sides by \(-4\), we get \(q + r = 11\), so \(q = 11 - r\). Now substitute \(q = 11 - r\) into the first equation \(12q+3r = 15\). So \(12(11 - r)+3r = 15\). Expand: \(132-12r + 3r = 15\), combine like terms: \(132-9r = 15\), then \(-9r=15 - 132=-117\), so \(r=\frac{-117}{-9}=13\). Then \(q = 11 - 13=-2\). So the solution is \((-2,13)\). Wait, but let's check the options. The first option is \((-2,13)\). Wait, but when I substitute \((-2,13)\) into the first equation: \(12(-2)+313=-24 + 39 = 15\), which matches. Substitute into the second equation: \(-4(-2)-413 = 8 - 52=-44\), which also matches. Wait, but earlier when I thought I made a mistake, but it's correct. Wait, but let's check the other options. Wait, maybe I miscalculated before. Let's check option \((8,-1)\): First equation: \(128+3(-1)=96 - 3 = 93
eq15\). Option \((-18,29)\): \(12(-18)+329=-216 + 87=-129
eq15\). Option \((15,-44)\): \(1215+3(-44)=180 - 132 = 48
eq15\). So the correct solution is \((-2,13)\). Wait, but when I first solved, I thought I made a mistake, but no, it's correct.

Answer:

\((-2, 13)\) (corresponding to the option \((-2, 13)\))