Question

solve the system by substitution. check your solution.

1. $4x + 5y = 7$

$5y = 2x + 19$

1. $2x = 4y - 5$

$2x - 3y = 1$

Explanation:

Problem 6

Step1: Substitute \(5y\) in first equation

We have \(4x + 5y = 7\) and \(5y = 2x + 19\). Substitute \(5y = 2x + 19\) into \(4x + 5y = 7\), we get \(4x+(2x + 19)=7\).

Step2: Solve for \(x\)

Simplify the equation: \(4x+2x + 19=7\), combine like terms: \(6x+19 = 7\), subtract 19 from both sides: \(6x=7 - 19=-12\), divide both sides by 6: \(x=\frac{-12}{6}=-2\).

Step3: Solve for \(y\)

Substitute \(x = - 2\) into \(5y=2x + 19\), we get \(5y=2\times(-2)+19=-4 + 19 = 15\), divide both sides by 5: \(y=\frac{15}{5}=3\).

Step4: Check the solution

Substitute \(x=-2\) and \(y = 3\) into \(4x + 5y\): \(4\times(-2)+5\times3=-8 + 15 = 7\) (matches the first equation). Substitute into \(5y\) and \(2x + 19\): \(5\times3 = 15\) and \(2\times(-2)+19=-4 + 19 = 15\) (matches the second equation).

Step1: Substitute \(2x\) in second equation

We have \(2x = 4y-5\) and \(2x-3y = 1\). Substitute \(2x = 4y - 5\) into \(2x-3y = 1\), we get \((4y-5)-3y=1\).

Step2: Solve for \(y\)

Simplify the equation: \(4y-5-3y = 1\), combine like terms: \(y-5 = 1\), add 5 to both sides: \(y=1 + 5=6\).

Step3: Solve for \(x\)

Substitute \(y = 6\) into \(2x=4y-5\), we get \(2x=4\times6-5=24 - 5 = 19\), divide both sides by 2: \(x=\frac{19}{2}=9.5\).

Step4: Check the solution

Substitute \(x = 9.5\) and \(y = 6\) into \(2x\): \(2\times9.5 = 19\) and \(4y-5=4\times6-5 = 19\) (matches the first equation). Substitute into \(2x-3y\): \(19-3\times6=19 - 18 = 1\) (matches the second equation).

Answer:

\(x=-2\), \(y = 3\)

Problem 7