What is the solution of [tex]$3 x+5=2 x-7 ?$[/tex]
A. the [tex]$x$[/tex]-coordinates of the intersection point of the lines [tex]$y=3 x+5$[/tex] and [tex]$y=2 x-7$[/tex]
B. the [tex]$x$[/tex]-coordinates of the [tex]$x$[/tex]-intercepts of the lines [tex]$y=3 x+5$[/tex] and [tex]$y=2 x-7$[/tex]
C. the [tex]$y$[/tex]-coordinate of the intersection point of the lines [tex]$y=3 x+5$[/tex] and [tex]$y=2 x-7$[/tex]
D. the [tex]$y$[/tex]-coordinates of the [tex]$y$[/tex]-intercepts of the lines [tex]$y=3 x+5$[/tex] and [tex]$y=2 x+7$[/tex]

Solve the linear equation 3 x + 5 = 2 x − 7 to find x = − 12 .
Determine the intersection point of the lines y = 3 x + 5 and y = 2 x − 7 by solving the system of equations, resulting in the point ( − 12 , − 31 ) .
Calculate the x-intercepts of the lines y = 3 x + 5 and y = 2 x − 7 , which are − 3 5 ​ and 2 7 ​ , respectively.
Find the y-intercepts of the lines y = 3 x + 5 and y = 2 x − 7 , which are 5 and − 7 , respectively. The solution to the equation is − 12 ​ .

Explanation

Problem Analysis We are given the equation 3 x + 5 = 2 x − 7 and asked to find its solution. We are also asked to find the x-coordinate of the intersection point of the lines y = 3 x + 5 and y = 2 x − 7 , the x-coordinates of the x-intercepts of the lines y = 3 x + 5 and y = 2 x − 7 , the y-coordinate of the intersection point of the lines y = 3 x + 5 and y = 2 x − 7 , and the y-coordinates of the y-intercepts of the lines y = 3 x + 5 and y = 2 x − 7 .

Solving the Equation First, let's solve the equation 3 x + 5 = 2 x − 7 for x .
Subtract 2 x from both sides: 3 x − 2 x + 5 = 2 x − 2 x − 7 , which simplifies to x + 5 = − 7 .
Subtract 5 from both sides: x + 5 − 5 = − 7 − 5 , which simplifies to x = − 12 .

Finding the Intersection Point The x-coordinate of the intersection point of the lines y = 3 x + 5 and y = 2 x − 7 is the same as the solution to the equation 3 x + 5 = 2 x − 7 , which we found to be x = − 12 . To find the y-coordinate of the intersection point, we can substitute x = − 12 into either equation. Using y = 3 x + 5 , we get y = 3 ( − 12 ) + 5 = − 36 + 5 = − 31 .

Finding the x-intercepts To find the x-intercept of the line y = 3 x + 5 , we set y = 0 and solve for x : 0 = 3 x + 5 . Subtracting 5 from both sides gives 3 x = − 5 , so x = − 3 5 ​ .
To find the x-intercept of the line y = 2 x − 7 , we set y = 0 and solve for x : 0 = 2 x − 7 . Adding 7 to both sides gives 2 x = 7 , so x = 2 7 ​ .

Finding the y-intercepts To find the y-intercept of the line y = 3 x + 5 , we set x = 0 and solve for y : y = 3 ( 0 ) + 5 = 5 .
To find the y-intercept of the line y = 2 x − 7 , we set x = 0 and solve for y : y = 2 ( 0 ) − 7 = − 7 .

Final Answer Therefore, the solution to the equation 3 x + 5 = 2 x − 7 is x = − 12 . The x-coordinate of the intersection point of the lines y = 3 x + 5 and y = 2 x − 7 is − 12 , and the y-coordinate is − 31 . The x-intercepts of the lines y = 3 x + 5 and y = 2 x − 7 are − 3 5 ​ and 2 7 ​ , respectively. The y-intercepts of the lines y = 3 x + 5 and y = 2 x − 7 are 5 and − 7 , respectively.


Examples
Understanding linear equations and their intersections is crucial in various fields. For instance, in economics, the intersection of supply and demand curves represents the market equilibrium point. Similarly, in physics, analyzing the motion of objects often involves solving linear equations to determine positions and velocities at specific times. These concepts are also fundamental in computer graphics for rendering lines and shapes on the screen.

Answered by GinnyAnswer | 2025-07-06