Answer:
[tex](2,5)[/tex]
Step-by-step explanation:
[tex]5x+3y=25\\-3x+5y=19[/tex]
To solve this system of equations, we can first start by solving for one of the two variables in one of the equations. Let's use the first equation to solve for our [tex]x[/tex] value:
[tex]5x+3y=25[/tex]
Subtract [tex]3y[/tex] from both sides of the equation:
[tex]5x=25-3y[/tex]
Divide both sides of the equation by the coefficient of [tex]x[/tex], which is [tex]5[/tex]:
[tex]x=5-\frac{3y}{5}[/tex]
Now that we have our [tex]x[/tex] value, we can substitute it into the second equation to solve for our [tex]y[/tex] value:
[tex]-3x+5y=19[/tex]
Substitute:
[tex]-3(5-\frac{3y}{5})+5y=19[/tex]
Distribute the [tex]-3[/tex] into the parentheses:
[tex]-15+\frac{9y}{5} +5y=19[/tex]
Combine the fractions by achieving a common denominator between [tex]\frac{9y}{5}[/tex] and [tex]5y[/tex]. (Multiply [tex]\frac{5y}{1}[/tex] by [tex]\frac{5}{5}[/tex]):
[tex]\frac{5y}{1}[/tex] × [tex]\frac{5}{5}[/tex]
[tex]=\frac{25y}{5}[/tex]
[tex]\frac{25y}{5} +\frac{9y}{5}[/tex]
=[tex]\frac{34y}{5}[/tex]
The equation now looks like:
[tex]-15+\frac{34y}{5} =19[/tex]
Multiply both sides of the equation by [tex]5[/tex] to get rid of the fraction:
[tex]-75+34y=95[/tex]
Add [tex]75[/tex] to both sides of the equation:
[tex]34y=170[/tex]
Divide both sides of the equation by the coefficient of [tex]y[/tex], which is [tex]34[/tex]:
[tex]y=5[/tex]
Now with our [tex]y[/tex] value, we can substitute into the equation representative of the [tex]x[/tex] value:
[tex]x=5-\frac{3y}{5}[/tex]
Substitute:
[tex]x=5-\frac{3(5)}{5}[/tex]
Multiply:
[tex]x=5-\frac{15}{5}[/tex]
Divide:
[tex]x=5-3[/tex]
Subtract:
[tex]x=2[/tex]
Therefore, the two roads intersect at the position [tex](2,5)[/tex].
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You can check your work by substituting the solved variable values into the initial system of equations:
[tex]5x+3y=25\\-3x+5y=19[/tex]
[tex]5(2)+3(5)=25\\-3(2)+5(5)=19[/tex]
[tex]25=25\\19=19[/tex]
Since the sides of the equations are equal to each other, our solution is correct!
