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Sagot :
To solve the system of equations and express each equation in slope-intercept form, let's go through the steps in detail:
System of Equations:
[tex]\[ \begin{aligned} 2x + y & = 3 \\ -2y & = 14 - 6x \end{aligned} \][/tex]
### Step 1: Convert the first equation to slope-intercept form (y = mx + b).
Equation: [tex]\( 2x + y = 3 \)[/tex]
1. Isolate [tex]\( y \)[/tex].
[tex]\[ y = 3 - 2x \][/tex]
2. Rearrange to match the slope-intercept form [tex]\( y = mx + b \)[/tex].
[tex]\[ y = -2x + 3 \][/tex]
Slope-intercept form of the first equation:
[tex]\[ y = -2x + 3 \][/tex]
### Step 2: Convert the second equation to slope-intercept form (y = mx + b).
Equation: [tex]\( -2y = 14 - 6x \)[/tex]
1. Divide all terms by -2 to isolate [tex]\( y \)[/tex].
[tex]\[ y = \frac{14 - 6x}{-2} \][/tex]
2. Simplify the expression on the right-hand side.
[tex]\[ y = -\frac{14}{2} + \frac{6x}{2} \][/tex]
[tex]\[ y = -7 + 3x \][/tex]
3. Rearrange to match the slope-intercept form [tex]\( y = mx + b \)[/tex].
[tex]\[ y = 3x - 7 \][/tex]
Slope-intercept form of the second equation:
[tex]\[ y = 3x - 7 \][/tex]
### Step 3: Solve the system of equations.
We have the following two equations in slope-intercept form:
[tex]\[ \begin{aligned} y &= -2x + 3 \\ y &= 3x - 7 \end{aligned} \][/tex]
To find the point of intersection (solution), set the equations equal to each other:
[tex]\[ -2x + 3 = 3x - 7 \][/tex]
1. Combine like terms and solve for [tex]\( x \)[/tex]:
[tex]\[ 3 + 7 = 3x + 2x \][/tex]
[tex]\[ 10 = 5x \][/tex]
[tex]\[ x = 2 \][/tex]
2. Substitute [tex]\( x = 2 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. Using [tex]\( y = -2x + 3 \)[/tex]:
[tex]\[ y = -2(2) + 3 \][/tex]
[tex]\[ y = -4 + 3 \][/tex]
[tex]\[ y = -1 \][/tex]
Solution of the system:
[tex]\[ x = 2, \; y = -1 \][/tex]
### Summary:
1. First equation in slope-intercept form:
[tex]\[ y = -2x + 3 \][/tex]
2. Second equation in slope-intercept form:
[tex]\[ y = 3x - 7 \][/tex]
3. Solution to the system:
[tex]\[ (x, y) = (2, -1) \][/tex]
Thus, the equations in slope-intercept form are:
[tex]\[ \begin{array}{l} y = -2x + 3 \\ y = 3x - 7 \end{array} \][/tex]
And the solution to the system is:
[tex]\[ (x, y) = (2, -1) \][/tex]
System of Equations:
[tex]\[ \begin{aligned} 2x + y & = 3 \\ -2y & = 14 - 6x \end{aligned} \][/tex]
### Step 1: Convert the first equation to slope-intercept form (y = mx + b).
Equation: [tex]\( 2x + y = 3 \)[/tex]
1. Isolate [tex]\( y \)[/tex].
[tex]\[ y = 3 - 2x \][/tex]
2. Rearrange to match the slope-intercept form [tex]\( y = mx + b \)[/tex].
[tex]\[ y = -2x + 3 \][/tex]
Slope-intercept form of the first equation:
[tex]\[ y = -2x + 3 \][/tex]
### Step 2: Convert the second equation to slope-intercept form (y = mx + b).
Equation: [tex]\( -2y = 14 - 6x \)[/tex]
1. Divide all terms by -2 to isolate [tex]\( y \)[/tex].
[tex]\[ y = \frac{14 - 6x}{-2} \][/tex]
2. Simplify the expression on the right-hand side.
[tex]\[ y = -\frac{14}{2} + \frac{6x}{2} \][/tex]
[tex]\[ y = -7 + 3x \][/tex]
3. Rearrange to match the slope-intercept form [tex]\( y = mx + b \)[/tex].
[tex]\[ y = 3x - 7 \][/tex]
Slope-intercept form of the second equation:
[tex]\[ y = 3x - 7 \][/tex]
### Step 3: Solve the system of equations.
We have the following two equations in slope-intercept form:
[tex]\[ \begin{aligned} y &= -2x + 3 \\ y &= 3x - 7 \end{aligned} \][/tex]
To find the point of intersection (solution), set the equations equal to each other:
[tex]\[ -2x + 3 = 3x - 7 \][/tex]
1. Combine like terms and solve for [tex]\( x \)[/tex]:
[tex]\[ 3 + 7 = 3x + 2x \][/tex]
[tex]\[ 10 = 5x \][/tex]
[tex]\[ x = 2 \][/tex]
2. Substitute [tex]\( x = 2 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. Using [tex]\( y = -2x + 3 \)[/tex]:
[tex]\[ y = -2(2) + 3 \][/tex]
[tex]\[ y = -4 + 3 \][/tex]
[tex]\[ y = -1 \][/tex]
Solution of the system:
[tex]\[ x = 2, \; y = -1 \][/tex]
### Summary:
1. First equation in slope-intercept form:
[tex]\[ y = -2x + 3 \][/tex]
2. Second equation in slope-intercept form:
[tex]\[ y = 3x - 7 \][/tex]
3. Solution to the system:
[tex]\[ (x, y) = (2, -1) \][/tex]
Thus, the equations in slope-intercept form are:
[tex]\[ \begin{array}{l} y = -2x + 3 \\ y = 3x - 7 \end{array} \][/tex]
And the solution to the system is:
[tex]\[ (x, y) = (2, -1) \][/tex]
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