4.) $9 x+5=3 x-19$
5.) $x+8=13+2 x$
6.) $14 x=18+5 x$

Solve the first equation 9 x + 5 = 3 x − 19 by isolating x : x = − 4 .
Solve the second equation x + 8 = 13 + 2 x by isolating x : x = − 5 .
Solve the third equation 14 x = 18 + 5 x by isolating x : x = 2 .
The solutions are x = − 4 , − 5 , 2 ​ .

Explanation

Problem Analysis We are given three linear equations and our goal is to solve each equation for the variable x . We will use algebraic manipulation to isolate x on one side of each equation.

Solving Equation 4 For the first equation, 9 x + 5 = 3 x − 19 , we want to isolate x . First, subtract 3 x from both sides of the equation: 9 x − 3 x + 5 = 3 x − 3 x − 19 This simplifies to: 6 x + 5 = − 19 Next, subtract 5 from both sides: 6 x + 5 − 5 = − 19 − 5 This simplifies to: 6 x = − 24 Finally, divide both sides by 6: x = 6 − 24 ​ = − 4

Solving Equation 5 For the second equation, x + 8 = 13 + 2 x , we again want to isolate x . First, subtract x from both sides: x − x + 8 = 13 + 2 x − x This simplifies to: 8 = 13 + x Next, subtract 13 from both sides: 8 − 13 = 13 − 13 + x This simplifies to: − 5 = x So, x = − 5 .

Solving Equation 6 For the third equation, 14 x = 18 + 5 x , we isolate x . First, subtract 5 x from both sides: 14 x − 5 x = 18 + 5 x − 5 x This simplifies to: 9 x = 18 Finally, divide both sides by 9: x = 9 18 ​ = 2

Final Answer Therefore, the solutions to the three equations are: Equation 4: x = − 4 Equation 5: x = − 5 Equation 6: x = 2


Examples
Linear equations are used in various real-life scenarios, such as calculating the cost of items, determining distances, and converting units. For example, if you know the total cost of a taxi ride is $10 and there is a fixed charge of $2, you can use a linear equation to find the cost per mile. If x is the number of miles, the equation would be 2 + a x = 10 , where a is the cost per mile. Solving this equation gives you the cost per mile, which is a practical application of linear equations in everyday life.

Answered by GinnyAnswer | 2025-07-08